Let's see. So jade today says when is NFA next week not financial advice with me and guy from coin Bureau and Ben from the crypto worse. It should be every Thursday It's just that last couple times We've had some some things going on guy had to do something in the metaverse some kind of deal with finance and Like a shark tank type thing sounds interesting But it was on the exact same day and time. So that's done. So we're gonna be it should be every Thursday at 9 a.m. Eastern time and it'll be on guys channel and coin Bureau clips next week Yeah, it's probably true Rob. I think you can hammer out a lot of these crypto regulation issues If you just sit down with Gary at a cafe over some Cricket powder tea. I don't I probably it's it's probably very simple. I got to tell you between us It's no one's it's just us right now Nobody's watching I think that they're dragging their feet on purpose Call me crazy, but I'm pretty sure that they could make this all very clear very quickly and Just move forward, but it's like for some reason they don't want to have clear regulation and that's it No, that's nice Well, welcome Vicki says I have a new interview channel. I will let us interview in a month. Absolutely You know how to get a hold of me? Yeah, that's right. Hello tnd Do you hold chain like yes, I do And uh There was this great article Let me see if I can pull it up. Did you guys know? Let me show you this It's a couple things I want to show you This dynamic NFTs so chain link is This is updated on january 23rd. So that's already it's coming into fruition, but a dynamic NFT. What is that? Well, NFTs are pretty static, you know, they don't really change too much I mean upgrade a bowl and stuff like that But you mean you have to do that manually But I don't this is all metaverse stuff. So like it would be like Or like later in game like if you go into a certain section of a game and you conquer some boss Automatically you level up and then your NFT levels up and it's like an automatic type of thing Which is on chain stuff. But what about off chain and I thought it was interesting So chain link, of course is an oracle and it brings an outside data into On into the blockchain, right? So let's just say that in the future We have an NFT that represents a house. Well, how do you know what that house? What the value is what's happened to it in the past good and bad? And everything kind of like information that you want to get especially if it's on an NFT So what it does is it'll update it automatically without having put things in it's kind of like Think of like Carfax, but on the blockchain and it's for houses and it could be for cars And it could be for any kind of asset that you really have and you could just see like all the data On the blockchain for anybody And so like let's say like in this example, let's say it replaces the roof You're like, well, you know, that's good to know and and then of course We can we can put that in and take a look at comparables other houses as far as like new roofs in a certain section And and what is that the price per square footage or let's say that they did new doors new new windows New something or and also let's say that there was a massive foundation crack And they had to fix that and it would all be in the NFT data. So why is that important? I think it's important because like if we want to use these types of things In the future it makes our lives a heck of a lot easier to do. So if you're an assessor And you're trying to look at or a real estate agent You're trying to look at like like give me these houses and these properties and and this type of price and there's no, you know Wacky information that that comes out on closing. Well, this will be where it is I know some people will say well, that seems kind of I mean, don't we have people to do that? Yes, we do we do all those things and of course That's the whole points of technology is to make things a lot faster easier simpler and cheaper and cutting out middleman If we can do that that will be fantastic. So to answer your question I do own chain like and especially it's interesting that they're doing these types of things. So, yeah Interesting also, I wanted to talk like that story, you know, just took up a whole lot of a lot of timeframe, but Ah, hold on. Let's see here I'll answer this What's your opinion of the Celsius new co-plan? It sounds like an awful plan with retail clawbacks And those are the 5k on the platform get half back compared to those with less than 5k So people have been like asking me to cover The Celsius plan what things are going on with Celsius and how's this thing with Celsius? and I haven't done much of it because First of all, uh, so everybody knows, you know, I promoted Celsius I talked about it quite a bit as a matter of fact And I have six figures stuck on that platform So I'm right with you on on Celsius and Celsius was it was a project that worked until it didn't And it's the same thing with Voyager Voyager worked until different when tell it didn't until they made this awful mistake of Loaning 640 million to three rows capital uncollateralized geniuses So this with this Celsius plan To cover it I don't think I can cover it But it's a lot of minutiae and the back and forth And it seems like the only people that know what's really going on to me and my What I see is Simon Dixon. So follow him Aaron Bennett Tiffany Fong And Cam Cruz These four people know exactly what's going on and you can follow others and you know to see what's happening But they they really keep up with the details of what's happening I was asked to get to cover it when there was this thing called a Celsius squeeze going on And I sat in on on a twitter space is live and I listened to everybody Complain and throw mud back and forth and nothing got accomplished whatsoever And now those same people the Celsius squeezers are talking about how great Celsius still is I mean some of them are there was a report that came out that basically said that Alex Mishinsky and Celsius was a Ponzi and they said we can't say, you know The exact words but it operated as a Ponzi and these people looked at that and go wow It's fantastic. They didn't call it a Ponzi. I'm like, did you not read the report? so The things that are going on and there's like I forgot the name terrible for something like that is is coming in They're like, we're gonna, you know make this thing great and it's gonna be awesome and And I'll let you know if that is really what it comes down to it, but right now There's really in my personal opinion It's not like there's a huge amount and people would say but Rob what about these? Retail clawbacks and things like that. Well, look we try to fight Kirkland and Ellis for them to make The crypto that we had on that platform to not allow them to classify that As Celsius's property and that failed miserably and they were able to do that And then for everybody who has it in you know and loans and custody and of course we can fight them But really what it's going to come down to is just time and what happens me personally I would just rather liquidate And just move on because I have a much better feeling that I can Use my funds a lot better than a new company coming in and saying, you know, we're gonna do this great stuff and this great stuff I don't trust anybody involved with it and I'll go out of my way to uh to make sure I didn't really want to talk about it, honestly That's just it. I mean why can bring some people on I asked Simon to come on to probably be here next week We'll talk all about it quite honestly I think it's uh, I'm just kissing that six figures. Goodbye so all right Do you think Celsius plan will not I don't know because Oh, first of all this new place. They're like, we're not going to be They they they say it's funny because like they say that it's not going to do like the old ways and they're not going to bring in the old Uh management team and it's going to be totally new and it's going to be totally awesome Basically what it is is they're getting they're getting your assets on the cheap And they're going to fire up this bitcoin mining platform and everything's going to be fantastic. Look Maybe it will I just reserve judgment, but I mean from from here on out Like I'm just tired of people saying like you should get behind Celsius and those and like Celsius is a bunch Of fraudsters and and people would say well, Rob, you don't understand They're doing this this great bitcoin mining operation and then if you take a look at All the financials you'll see that they were hemorrhaging money left and right They were actually losing money This bitcoin mining operation when they should have been making money So if this new company comes in they're like, oh, this is going to be great. I'll reserve judgment. So that's it I have no faith in ever anybody who is around that project The Yeah, follower and ben at Tiffany Tampong They're great Please tell us citizens when we might get sweat coin probably never No, it's gonna be let's see gary genzler gets out of office in 2026 probably around then So I don't know man.  I don't know and like I said, I mean Simon Dixon has much more faith in in in this new company than I do So maybe he can talk some sense into me, but I just don't Yeah, I just don't You know what it would be great is if they do liquidate What's what's people like I lost my mind. It's awful Well, you know what? When they liquidate and all these things happen Now you can actually claim this on taxes as massive losses So like right now, you know, you can't because everything's up in the air in chapter 11 They could come out And it could be this or could be that I mean talk to your cpa. I'm not a cpa But uh, I mean, I just rather write it off and I can have massive losses and I can carry those over for years to come. So I don't see I don't see it Thank you. Jeff. We all got played moving on Let's see I think that's it. There was a couple things that um To answer there's a couple questions. Yes. I'm still dcaing. I still I still have big plans. I think we're gonna see I mean this this year is going to be pretty rocky. We talked about this yesterday and on uh, NFA But uh 2024 run the bitcoin having I think we'll start to see more a little more stability. Hopefully if we don't go and go into a major recession But I think like here's a here's the thing if we go into a recession I think it'll be i'm i'm hoping q3 or q4 this year And the thing about recessions they last about a year, right? Depending maybe 14 months maybe 10 months about a about a year or so and it's a natural progression So if we extrapolate that out Then where are we? Well, we're in towards the end of 2024 After the bitcoin having and what happens after the bitcoin having usually we see uh, some major price increases and a little bit more of like a Stare step action not that it goes straight up, but it looks a lot better So I think it's just perfect timing. I'm just hoping It works out like that if we don't get a recession we don't get a recession, right? Yeah, to me it doesn't really matter too much because I just put my money away I dollar cost hours and I take a look at some factors of what I'm going to sell There's a link in the description of the video of when I'm going to sell 80 percent of my crypto and I take a look at uh multiple of of data points and you can Put that into your plan or not Do whatever you want to do So Noah says are you worried at all about itrust's custody program? So we had we did an interview um From itrust for their new custody program and they're moving so the so the thing was is that They're still they were using coin-based custody and fire blocks and the person that set that up Was their custody provider and then the new one Uh, I shoot what was the name scaling not scale fortress and fortress they're more of a Not a better But they use the same they they use fire blocks for custody service and a couple of the different ones and it'll allow us to like do Not anything fantastic. Just a different way. They want to they want to move or they want to grow So for me as far as like the custody thing the big thing that I liked about that Was the documentation with the terms and conditions which talked about they don't intermingle funds and we took a look at Okb okx the crypto exchange They just were blowing up a couple of weeks ago and I was like I don't understand why this is going on Because in terms of conditions it basically said we can co-mingle your funds and we can label it as okx's funds Watch the video. I'll show you so for fortress And for itrust they said they say totally different like we keep everything separate It is in your name under your account If something happens with us we just switch custody providers and that's pretty much it So am I concerned? I'm not uh, I'm not uh like A hundred percent sleep easy. There's always a concern everywhere. There's always risk for every type of investment you do Great thing about uh, but I trust and Roth IRAs is that you can't put You know, you can't take your life savings and sell your house and your kidneys and your kids and go Okay, and put it all on black and then uh, you know gambling it away like essentially what you could do Well, a casino sir or actually any stock that's out there. So for me I'm not as concerned with it as uh I say like these d5 plays or these centralized exchanges That's it So meme says but in pr if you don't pay capital gains under the grant How can we have capital losses? So I moved here in 2021 and the majority of my crypto was purchased in 20 because I dollar cost average in 2018 2019 And well actually every single year. I didn't sell all my crypto So for my cpa what they tell me is even if you moved here The crypto that you had before It doesn't automatically hit like that the day that you move here and become a resident and file for uh x 60 It needs to be 21 and 22 Moving from that point forward You won't have any losses as far as like not losses all right capital gains But to say that you accumulated beforehand that's still Uh detrimental to you because you're gonna pay capital gains in that And that's how it goes. So that's it So for I guess that would be for me for everybody else who's not in portorico It's a little bit different. You have capital gains to pay Cocchino says how long do you plan to stay in portorico? We Well, we'll live here our entire lives But uh, we'll be traveling Yeah, we actually go back to texas end of march. So you have to go back there for business. So Beardy says rob can get a wound back lesson, you know, that's all I did for five years I just Instructed nurses practitioners and surgeons how to use the wound back I it's been a while, but I could still do it Uh Let's see rob. What are you buying at the moment as much as I can I got a bitcoin ethereum the blue chips Cardano chain link near polka dot Avalanche I'm always forgetting some and some other ones and there's a couple of risky ones. I don't talk about but yeah, that's it So mtb says what do you think of blur? So blur just did There was an act not an acquisition, but they're like They're the nft platform that is Kind of going against open c And their big thing is correct me if i'm wrong is not about The royalties for the nft creators They tried to cut that down or cut that out and that went against what Open c did and they were doing so well I think that the volume of nft that was uh nfts are being bought Rivaled that of open c and now you got more of blur Now blur did a big airdrop and a lot of people that were buying and selling nfts got a bunch of tokens So looks good me personally. I don't really get into nfts right now So it's not a big big thing, but for me it just kind of comes down to this people will always use a platform that is Faster and cheaper and really cheap. That's what it was down to as long as the as the quality is somewhat there So who knows blur might be the next Big open c. I don't know So noa says cool My only concern is that with the crypto is not subject to security regulations What insurance we have that our crypto is safe in the event fortress goes bankrupt. So again And you can look in the terms conditions At fortress and also what I trust and you should also reach out to the the folks that I trust and ask them the specific question because i'm just a guy Who is talking to his computer? on a really decent That pretty good green screen And uh again the video we talked about it said quite specifically No co-mingling of funds and again There could be other other parts of that that i'm not privy to but that's what I saw in the terms of conditions And if there's not a co-mingling of funds That means that your account and your crypto doesn't get mingled in with everybody else's so they can't use your funds to buy condos In the Bahamas like sam bakeman freed to the ftx people That's it James says stay with the one true crypto. I gotta tell you I should just do that. It's much easier I never defy was always a like everybody said it's like the next big thing It's gonna be awesome and it very well could be but there's a lot of bumps to get there So I don't know me personally. I just Like I don't know where things are going, but that's why I like layer one solutions. That's why I like you know, that's why you hear like ethereum and avalanche and cardano and poke it out chain link solutions because like what's built on ethereum Darn near everything is built on ethereum If you take a look at how many years you 20 tokens there are is a lot So does that mean it's gonna keep being that way? Who knows? But uh, you've got other players in there. That's why I do those things So even if like So like people build them a theorem they do defy projects or they do nft So they do play their own gaming. It's all built on one thing right layer one solution So you can try to pick those winners, but it's tough Yeah, who bring up blood? No, that's right Yeah, fame mma Let's see you better load up on mattock before zk evm release You know, somebody said like steven over at uh, San Juan smokehouse said he believes that uh, because coinbase They're working on uh on the optimism platform and they released this layer two solution called base He said that's gonna kill Polygon and I was like, I don't know because I mean Coinbase also came out with their nft marketplace. I don't know how good that's doing, but I don't really hear too much about it so Who knows Not for sure, but you know, polygons got so many so many partnerships and so many things are I mean It really Does pretty well at the gaming sector. That's for sure. So Who knows? I don't know Is cardano still a ghost chain? Nah There's a link in the description. It's called cardano ghost chain. It's a slideshow it shows you I don't think Cardano is a ghost chain per se onwards and operates Yeah, johnny says Please try to people trying to play the blur Farming game need to be aware that they are playing against 150 people who are controlling the platform So do your due diligence before bidding Yeah, that sounds risky to me Like, you know, think about it this way. I don't know how risky people are and people are some are risky Some are not just remember this like you know, there's great gains to be had so First of all, does anybody gamble here like in a casino? Raise your hand. I've done it before Yeah, and uh, I think Most people don't take their entire life savings and And gamble the majority right will say 95% right, but there's that 1 to 5% of degenerates who just cannot stop gambling. That's just how it is. So like For these these plays I think the majority of people I think I'm not sure are not degenerates. They just kind of want to like well, maybe I'll make a little money and I'll you know, I'll take 20 bucks or 100 bucks or $10,000 or whatever Whatever is not big money to you and just kind of play the game and see what you do I think there's no problem with that, right? No problems with that I just have a problem when people like they're sitting there and they're they're They're gambling on leverage And they're going I can make it and they can't pay bills. That's the problem. But again, not your dad So you got to make that decision for yourself Now J. H says well Rob's only my dad. No one else What he does isn't what you should do. Don't trust anyone after sell fdx, etc. Exactly, right? Oh, that's pretty good.  They're fantastic at it.  I think it's gonna hopefully go away, but you're gonna hear a lot of it Diamond hands bro Robby looked into real world assets on chain rwa is the next big narrative and this is low-key White base launch with black rock in on it. That's interesting black rock and base Black rock and base. It'd be interesting Yeah, real world assets on on chain You know who wants who was doing a lot of that a couple years ago and still doing is is bosson or boson protocol b o s o n protocol they were they were So you could take your real world assets and put them in the metaverse and move back and forth, but we'll see BNB is a scam chain.  Well on the ads and things like that, of course Oh, uh real quick impromptu poll I don't do many Many live streams anymore Just don't so I might do some more later, but probably not But uh people like the conversation to to keep going and I think I by me not doing the live streams I kind of took that away from people so I have a question for everybody and lucas brought this to my attention What would be better for everybody? Telegram or discord To get a conversation going And the problem I've always had with telegram and discord is the amount of scammers and spammers there But what i'm gonna do is whichever one that we decide to launch on I'm just gonna the people who are wrenches Here will probably be the admins over there So we can do the same type of talking and they can protect people And that's the that would be the thing and then of course The link that you get to sign up will be private and uh There's a way to get it, but I first I had to figure out who's going to be is going to be telegram or discord Yeah, today's saturday. Can you believe this? There are some houses open. There's some houses open Yeah If only we could sell from the hard wall without fees, you know that but you know who's you know, who's making a killing Is the moon pays and the different plate and the different uh companies Wire or whatever else it is That allow you to to trade and sell in your wallet. They're making fat fee like five percent six percent four to six percent for sure like Visa doesn't even cost cost that much paypal doesn't cost that much, but those guys they're raking it. They're raking that money in So Yeah, let's see telegram discord discord discord discord i'll do a I'll do a poll like a real poll discord Discord to give me a wrench. Why does everybody like discord? I gotta ask you guys. Why do you like discord? Moon pay is awful. He says I have discord open all day every day No way salon is down. Huh? Yeah Easy to use and everyone uses it. Can you? Like I use discord on my on my uh on my um computer I don't really on my mac. I don't use it on the on the mobile device, but I really shouldn't start doing that Hmm. This goes a better layout less scammers on discord really uh Will Shaw says patreon now patreon is But then I got a charge Come on. Just save your money and put it into crypto bill bones. That's probably the thing about Celsius retail clawbacks bad news I mean, I will say like this like the clawbacks for uh the people that were A part of management they should do retail clawbacks I don't I don't think they should do but they probably will and again. I think this is again like Like people are like robbie should cover this you should cover this I just think it's gonna go really ugly for a while and uh that clawback will probably happen And what'll be interesting is that you know If that clawbacks if those clawbacks do happen, I'm pretty sure the the new company that tarah wolf, whatever it's called they're gonna Come out of that pretty happy because now they have all this crypto to play play with and uh good news for them They got added a massive discount.  Let's not claw back So that's all I need to know about that about them Maybe I'm wrong. I had a rough night Now this is good Ben and guy are both on telegram. That's true. I'm in ben's uh that little telegram group. It's quite interesting Sure, let's show So rob tell me to butt out if you know as if you could see your full-time job. We do other things I don't do anything so like I haven't worked for anybody since 2014 2015 somewhere around there And uh now like I got into online education And I was helping nursing students pass their clinical exam And this website's what allowed me to not Work for anybody else was pretty great. And then I got into real estate and that helped out a lot It was a more long-term rentals and things like that and then I That worked out pretty well from the money I got from the website and then did that and the money if we did that We got into uh an amazon business Amazon business is extremely boring, but it's quite profitable if you know what to do. It's very simple. You just Essentially reach out to distributors and say I'd like to give you a spreadsheet of the things that you have And I'll pick which ones I want to buy so like let's say you buy like a thousand 10,000 remote controls for an air conditioning unit and then you just profit off the difference You have them ship it to amazon warehouse and they sell it for you under your under your name And then uh, also sports facility in el paso. It's just all sand volleyball courts and uh Now we do more of uh the short-term rentals and air bnb. So This thing is that's why like I told everybody like on the next bull run When we hit the parabolic bull run and I that video in the link in the description Where I talk about how I'm going to sell like 80 of my crypto Once those parameters are hit. I'm stepping down And that's it for me.  Well, most people suck at selling and I think once it goes up and goes down people like Let me ask you guys this how many of you out there are like boy I wish I would have sold some more when bitcoin was at 69 000 and the three was almost 5000 and codona was three dollars Or has everybody sold the top?  I can do some other things. But yeah, like like these days. I feel like um I feel like I should do it So yeah, but that's it. All right, buddy coming up on an hour. I think that's good for a saturday. So look That's all we have for today. It was supposed to be an impromptu thing, but that's it So thanks so much for stopping by. I appreciate it time to go to the beach play a little volleyball That's a couple hours. That's it for me. So enjoy the rest of the day. We're over at adios


Freedom! Today I wanted to talk to you about, well, something I've already talked to you about. Communicating professionally. Communicating professionally is the very basis of being able to accomplish anything as a small YouTuber. If you can't take yourself seriously, and if you can't convey yourself in a way that others will take you seriously, then you're not going to catch many breaks. However, if you can communicate properly, write a good letter, forge a good email, speak properly, and Skype call, you can accomplish pretty much anything. Last time I talked about this, I was able to give away a SteelSeries Stratus, which was a gaming controller only compatible with iPhones, iPads, and iPhones. This time, unfortunately, I can't do that. However, what I can do is I can give away one of these. This is a Logitech H800. This is a wireless headset that I am able to give away to you, the audience, as a sponsorship here on Freedom. What that means is, whoever writes a very professional letter in the comments of this video, requesting a review unit, which would be this, has a chance of winning a sponsorship for this. So when you get it, you do an unboxing, a review, you tell us what you think, if you like it, how you're going to use it, that kind of thing. So really, that's all it takes to win. In the comments section down below, I want you to write me a nice letter. Keep it short, keep it to the point. Usually, if you're writing more than two paragraphs, it's too long. In addition, I want you to link one video that is meant to impress. This should be a standalone video. This should be something from your portfolio that Adam spoke about here. All decisions will be made within 48 hours of the release of this video. So take your time, write out something that you feel is going to impress on a professional level. Don't tell me your life story. Don't try to guilt me into picking you. Just leave it at the facts. How is it that you can help Logitech by getting this headset? Tell me in the comments section down below. I hope that this video has been helpful for you. If it has, let me know in the comments section down below. If you really like this video, hit the like button. Till next time, I'm Anthony with Freedom Life. Subscribe, comments please. Let's discover freedom. Click the first video to learn what is freedom and how it helps you grow. Click the second video to learn about sponsorships for all freedom partners, even if you have just 10 subscribers. And click the other videos like How Do I Leave Freedom to learn more? Click Get More Views to learn how to build your audience faster on YouTube. Click Get More CPM to learn how to make more money from your channel. And click Get More Music to learn where to get more music to spice up your videos. Click the first playlist to watch The George Show on Freedom. That's my show. Or watch The Anthony Show or The Alex Show also on Freedom.


Alright, so another rule concerning basic probabilities that we need to understand is the rule that concerns independent events. So definition of an independent event from a probability point of view is when the outcome of one event doesn't have any effect positive or negative or any sort on another event. So the standard example of that would be flipping coins. If I flip a coin, I either get heads or tails. If I then flip another coin, a different coin, or the same coin a second time, whatever I got on the first coin flip doesn't have any effect on what I get on the second flip. I'm not anymore or less likely to get heads on the second coin flip depending on whether I got a heads on the first coin flip. So for example, if I ask in that case if I flip two different coins and I want to know what's the probability that I get heads twice in a row, what's the probability that I get heads on the first coin flip and I get a heads on the second coin flip? The probability that I get heads on the first flip is one out of two. I have two options, heads or tails, and I've got a 50-50 chance of getting heads. I have to get heads on the first flip and I have to get heads on the second flip. So as May had seemed intuitive from the moment I asked the question, the odds of getting heads on both coin flips, the first one and the second one, is one out of four because I have 50-50 chance on the first one and a 50-50 chance on the second one. So only one of the four, I could have gotten heads heads, heads tails, tails heads, tails tails, only one of those four outcomes gave me heads on both of those coin flips. So that's an example of the more general rule for independent events, which is if I want to know what's the probability of A happening and some other event B happening, then I just take the two probabilities and multiply them together. As long as A and B are both independent events, they don't affect one another. So to work another example that involves independent events, let's say we do something with cards this time. I have an ordinary deck of 52 playing cards and I draw one card from that deck. Let's say I want to know what's the probability that I draw a queen of hearts out of the deck. So one way of thinking about that, of course, is there's only one queen of hearts and there's 52 cards. There's a one chance in 52 that I would draw a queen of hearts out of the deck. Or I could think of it as asking what's the probability that the card I draw is a queen and the card I drew is also a heart. So from that point of view, we can think about as two independent events. Whether I draw a queen or not doesn't have any bearing on whether it's a heart or not. I could also have thought about that problem as what's the probability that I drew a queen multiplied by the probability that I drew a heart. And there's four, well maybe the easier way to think about it is there's 13 different denominations of cards. Only one of them is a queen. So the chances are one out of 13 that I'm going to draw a queen. There's four different suits. Only one of them is hearts. So there's a one chance in four that I'm going to draw a heart. So if I multiply those two fractions together, one-thirteenth times one-fourth, then what I get is one-fifty-second. Again, there's more than one way to do the problem. Either way of thinking about it gives you the right answer, but it's certainly possible to think about this problem as a probability of two independent events because the two events are independent. So again, the basic rule here is whenever you have an AND problem, if you want one thing to happen and another thing to happen, what you do is you multiply the two probabilities together. That's distinct from the mutually exclusive events or problems where you add the two probabilities. But there's one very important caveat in this case. You can only multiply the probabilities for an AND event if the two events are independent. And this is more easily easy to trip over than the caveats for the other basic probability events because many times events are not independent. So it's always worth stopping and thinking about whether two events are in fact independent of one another. And to give you an example of how that can trip you up, I'll work one more example, which is to say, let's pick a random day of the year. So you can, for example, think of the day of your last birthday. So that's a calendar day on the year, and I don't know what that day that is. So from my point of view, I've just picked a random day by asking you what your birthday is. So since I don't know what your birthday is, if I ask, what's the probability that the date that I just randomly picked your last birthday was somewhere in June, July, August, or September. So it turns out that if we count up the days, that 122 of the days in the year are in June, July, August, and September. So there's a 122 out of 365 chance that the date we've now picked together has fallen in the range of June through September. I could also ask on that date of your last birthday, what's the probability that it was cold? If I define cold as if it got below freezing on that day, turns out at least at Clemson University, where this is being filmed, there's 62 days on average every year that drop below freezing. So the probability of that date that we're talking about being a cold day being below freezing, there's a 62 out of 365 chance that that randomly selected date was cold. If I now ask what's the probability that the date we've picked is in this range June through September and that it was cold, then I'll put a question mark here because of course this is going to turn out not to be true. I can say is that perhaps the probability of it being June through September multiplied by the probability that it was cold, in other words, is it 122 out of 365 multiplied by 62 out of 365? So that would work out to be if this is the right answer. So 122 out of 365, roughly one out of three days is in this summer time period, roughly one out of six is cold. If I multiply those two together, what I actually get is a number close to 5.7%. But of course you know that's not true. There's hardly any days in this range of June through September that are in fact below freezing. And in fact, if I look at the weather records, the same place that I got this record of cold days, there has never been a day in June through September in Clemson, South Carolina that was below freezing. So this is not the right answer. So this question mark doesn't hold. We can't multiply the two probabilities together. This is not the right answer. The correct answer to this question is zero. There's no chance that the date was both below freezing and in June through September, which we can only calculate with some extra information by going and looking at the weather records, for example. But the reason that this multiplying the two individual probabilities doesn't work in a case like this is not because this rule is incorrect. It's because this rule doesn't apply because the rule only applies if the two events are independent and whether a day is in the summer and whether a day is below freezing, those are not independent events. There's a correlation. So if there's a correlation between A and B, if A is correlated with B, or if knowing whether it's a date in the summer has some effect on whether the day is cold or not, then those events are correlated and we can't use this independent event formulation. So this one is just a reminder that while it's going to become very tempting anytime you see an and question, probably a something and something else, it's going to be very tempting to multiply two probabilities together. It's always worthwhile to stop and ask yourself, are these events independent? If so, then go ahead and multiply probabilities and the question will proceed fairly simply.


Listening to an 11-year-old tell me the way that also could have been how that's how that's the tone of the internet Anyway, it's an 11-year-old girl. So you actually heard at least it came from an 11-year-old girl instead of like so-and-so. It's canceled Thanks to the internet. It has never been easier to pile on to a public shaming in fact It's now one of America's favorite pastimes today.  No, he did not.


So, today we are going to talk of the consequences of special relativity. I am sure this portion is not new to you, but we will talk of the Doppler effect in light. The prerequisite for that is of course, you know a bit of time dilation things that we have covered earlier. I am sure you have heard of Doppler effect in sound. Let us see what it is there in light when we talk of electromagnetic waves, example light. And after that we will check or rather we will see whether the mass of a body actually whether it varies with velocity we may have heard of such things, but let us see where it comes from. So, let us first of all consider you know we consider a light source and then that is emitting light of a certain frequency. Let us just let us call that frequency new 0. And you have an observer here we show it by O here on to us right of your screen. I mentioned that of Doppler effect in sound. I am sure where instead of this light source you will be having someone some source which is emitting sound, which is making some noise and then emitting certain frequencies. And then it is all fine when they are at both of then both the observer and the source is at rest to teach other, but what happens if all of a sudden one of them starts moving. So, let us say the one of the source start moving and then the observer starts moving. Well, if you have the observer starts moving towards these the source of the sound do not you feel that you know the frequency of the sound that you are hearing has increased. It is more like a car coming towards you you know and it blows the car blows its horn. Then the more it comes towards you you find the pitch of the sound increasing. Well, so what is it with light? Do we observe certain the same things or is it a little bit different? Let us check that out. Of course, need to mention the very beginning. So, for propagation of light here you do not need a material medium because it can propagate to vacuum, but for sound of course you need a material medium. So, that is what we do we consider light source which is emitting light of a certain frequency that is new 0 and then we have an observer. Now, what we can also do you can also consider this a light source you know as a clock that is ticking like this tick tick tick and then at at each tick each one of this ticks and it is emitting a light wave. So, it is so what you are considering you are considering a light source as a clock in a sense that is ticking new 0 times per second and so on each tick it is emitting a wave. So, it is now why do we put all these clocks and things here then because if you put this clocks you know time dilation you really you can make the correlation and then we can use the all the principles of relativity that I have learnt earlier. And now let us see what happens if you have the source here you have the light source which is emitting a certain frequency and observer starts moving. Now, we can have three situations here either the observer is moving transverse your perpendicular to the direction of these light waves or it is moving towards a light source or it is moving away. So, let us let us consider each of them one by one ok. So, case number one. So, in this what we have is that the observer that is we denote by O here and it is moving uniformly with a certain velocity V transverse to the direction of the light waves. Now, if you are you know in the frame of the light source itself where the things are traced and what is the what is the proper time in the sense between the ticks you know you consider the light source to be a clock which is sticking and each of the stick it is emitting a it is a it is emitting a light wave. So, what is the proper time here it is called a T0 then how is it related with the frequency? Well you know you know frequency is in hertz and time it is in second. So, it is one by new 0 ok. So, that is the proper time interval between the ticks in the light source frame ok. Now, what about how would the observer then find this time to be ok. Now, in the frame of the observer the time elapsed between two ticks would be T. Now, T will not be same as T0 which was the proper time remember, but T will be T0 divided by root over of 1 minus V square by C square ok. It is this very simple time dilation formula that we have used here ok. Now, what would then be what would then be the frequency of the light as observed by the observer. Well you take just the inverse of the time here remember this time is the one measured by the observer in his or her frame ok. So, let us call this frequency as new T it is going to be 1 by T and then you know what 1 by T is you are going to you just see in the previous paragraph. So, you will see that it is root over of 1 minus V square by C square whole divided by T0. Now, T0 is also related with the frequency at which the light is being emitted ok. So, you see that. So, new T that is the frequency as measured by the observer that will be new 0 times root over of 1 minus V square by C square. Now, since V is always less than C here ok, it is going to be VT or the since the velocity of the of the observer is always going to be less than the velocity of light ok. So, what you find as new T that is the frequency as measured by the observer will always be less than the proper frequency ok proper time interval and then you know proper frequency if I can put it that way. So, new T here is less than new 0. So, that is the frequency that is that will be measured by an observer who is moving transversely to the to the source of light ok. So, that leaves us with two other cases one in which the observer is moving towards the towards the light source and in the other when the observer is moving away from the light source ok. So, let us consider the other one first as a namely the one in which the observer is moving away from the light source ok. So, here to you know what is the proper time between ticks it is going to be 1 by T 0 which is like 1 by new 0 that is when new 0 is the frequency at which the light is being emitted at the source ok. And remember here the observer is moving at the velocity V away from the light source. So, in the frame of the observer if T is the time that you know small t is the time that the observer will measure between each of the ticks. So, as to say then for in time T remember the observer is also moving with a certain uniform velocity V. So, the observer has already travelled V T that is that is the unit of distance here away from the source between the ticks ok. So, this implies that the time interval between two successive ticks as measured by the observer will be what will be V T by how much will it increase will increase by V T by C where T is the time as measured by you know by the observer here right. And V T by C and C is the velocity of light and that has the same you know that. So, the value of C is going to be the same in both frames ok. Now, what will be the total time between the arrival of successive waves or the successive light waves in the frame of the observer? It will be first the time between two ticks as measured by you know by the observer plus the amount of time you know the observer has moved away in a sense by which time the observer is already moved by a certain distance between these two ticks ok. So, the time lapse between the arrival of successive waves will be capital T which is small t plus V T by C now you simplify this as T plus and then you take a bracket of 1 plus V by C ok. Now, how is T related with the proper time small t that is small t is related with the proper time T 0 by small t being equal to T 0 by root over of 1 minus V square by C square and then you have the bracket 1 plus V by C. So, that simply simplifies to the one given at the bottom of your screen that is T 0 into root over of so, 1 plus V by C divided by 1 minus V by C. Now, how would you find the you know the frequency here? It is that all you need to take is the inverse of the time interval in this case. You see here I have put it as V nu minus. So, that is 1 by capital T. Now, that you just invert the expression that we see in the top of your screen and then you relate T 0 with the frequency that is the frequency with which the light is being emitted by the light source and then you immediately find what is a nu minus and then you can relate nu minus to nu 0 and then you see that nu minus is actually less than nu 0 here. So, in every case you see that the change in frequency so far we have seen that it is depending on the velocity of the observer, the velocity with which the observer is moving away from the light source here. So, what about the last case? So, in this case let us say the observer is moving towards the light source. So, by the same logic which we had applied earlier, the observer travels a distance which is V times small t towards the light source between successive ticks of the clock that we have considered here. So, the total time interval between the arrival of successive waves here, light waves here would be that is the capital T would be small t and then remember it is minus V t by c. Why minus? Because the observer is moving towards the light source and then we have kept the velocity of light is the same in both frames. I mean whether you are in a moving frame, uniformly moving frame or in a rest frame. So, that is why it is the same c. So, it is natural now. So, you do the simplification. You convert the small t into t0, t0 being the proper time. And here what we will find is that when you take the observed frequency, when you find the observed frequency, it is going to be, I have put V nu plus here. That is nu plus is 1 by capital T and we are going to see that nu plus here is more than nu 0. It is going to be more than the frequency of the emitted light. So, we have studied three cases in which the first case the observer was moving perpendicular to the light source. Then the one was moving away from the light source and then towards the light source. However, I should also mention here that in all cases you have seen that in all these cases the shift in the frequency as observed by the observer depends on the velocity of the observer here. So, you do see it in all this formula here. Another important point that I should mention here is that in each of the formulae that you have seen so far, I have always considered the light source to be at rest. Why? Because had I not for example, the light source was moving and then the observer was at rest. That would have been a similar situation because it is the relative velocity which is important here unlike the case of Doppler effect in sound. So, it is not only in the Doppler effect in sound, it is not the relative velocity. You have to take into consideration the velocity of the speed of the source and speed of the observer here. But here what we have it is the relative velocity of between the observer and the source which is going to be important here. So, if the source is moving or the observer is moving it does not matter as long because the velocity that we are considering here is actually the velocity V that you see in all the formulae will always be the relative velocity. So, having done this, let us spend a couple of minutes on where possible applications of this derivation would be. Well, as I told you, you see in the shift in frequency is related with the relative velocity with which the object is moving. So, that immediately told people that maybe it could be used to measure the speeds of distance Taylor objects like stars. So, what do people do? What do astronomers do in that case? What they do is that they take a photograph of the spectra of elements which is in the star. So, from the light which is emitted from the star the light spectra is analyzed. So, and where the frequencies are people find out and then they compare it with the known spectra of elements present in the star. So, these known spectra of elements you can find the spectra of elements by some experiment at the laboratory here on earth and then they compare that spectra with the spectra that was obtained from this star. Now, if the star is moving away or there is a relative velocity, there is a relative motion between earth and the star there and then the star we are going to have a shift in spectral frequency. Now, from the shift in spectral frequency you can then find the speed of the star. So, it is got lots of applications in astronomy of course, then many other some other discoveries were also based on this. Another application actually a very interesting application of this would be to measure the temperature of hot plasma in nuclear fusion experiments or the temperature of very hot gases. Now, what they do is that well the principle is almost similar you know to what I had what I have outlined earlier it is that they take the spectra of you know when things are moving at such high speed. So, they are emitting radiation and this radiation is analyzed and then if it is a known gas. So, from the known spectra you can find out what is the shift in the spectral frequency and then you can relate that with the speed of the molecule of the gas which is there inside your experiment and then if you relate if you remember you connect theory of gases. So, from this velocity or you know the mean velocity or the root mean square velocity you can always relate that to the temperature that can always be done from the kinetic theory. So, that will give an estimate of the temperature involved here. So, we have studied Doppler effect in light as against you know that now as against Doppler effect in sound and then we have also talked a bit about a few applications. Fine. So, now we are going to shift gear a little bit and do something a little bit different and go to another consequence of special relativity and study if the mass of a body actually varies with the velocity with which it is moving. When I know that this is one of the things that many of us are rather aware of when we study physics or any of the elementary courses of science we take mass increasing with velocity and equal to m c square but all of it in due time. To do this problem of where the mass of a body varies with velocity let us start with an example and then we will come to a conclusion. So, concentrate on the thing written down as s prime frame. So, what we have here is that we are considering the inelastic head on collision of two identical particles a and b in s prime frame let us say. So, what is this s prime frame? Well it is a frame which is let us say it is moving with a certain velocity certain uniform velocity v with respect to another frame s let us say. Now in it we have the inelastic collision of two bodies here named as a and b. So, they are of identical masses m prime both are of m prime and they are moving with the same speed you know opposite to each other and then they are heading towards a head on collision that means speed is u prime here. We have all put primes here because you know to keep in to keep you know in to make since we are talking of s prime frame. So, we are dealing with prime coordinates here let us put it that way and then all these velocities u prime you know they are moving parallel to the x axis. Fine so, they are identical masses also. Now how is the same event observed in s frame? Well you have different you have you know in the s frame you are going to see the same thing as some you know the velocities may not be the same. The velocities although they are moving the the speeds are all the same, but if you take the modulus of the speeds in s prime frame they are the same the directions may be different, but the speeds are the same. But then the velocities of the speeds are not going to be the same in s frame for a simple reason it is that you need to invoke Einstein's velocity addition formula to find what is the velocity in in s frame. And you go to see immediately that it is not going to be the same it is just written in you know in for your convenience at towards the end of the slide. For example, if you just consider u 1 so, of the mass of the velocity of mass 1 mass a rather you are going to see in terms of the s prime quantities and then the velocity v it is u prime plus v divided by 1 plus u prime v by c square. However, if it was minus u since was moving in other direction u 2 that is the other you know the other quantity that is minus u prime plus v into 1 minus u prime v by c square so, all going to be the same. So, you might be a little bit confused why you have written different masses for identical masses. So, in the s prime frame you had identical masses both are m prime. But why is it that they are different in in the s frame? Well at this stage I do not know why they are different rather I just take them to be different. So, just to keep myself in conjunction with the idea that I have different velocities I do not know what is going to the masses. So, if it is if my if my from my relativity from whatever I do it turns out from all this conservation laws I am going to invoke on if it turns out that these masses are the same they will turn out to be same m 1 will be equal to m 2. But right now I am going to take m 1 and m 2 different. Let us just see what happens. So, that is what we have we have the s prime frame and then in the s frame we are going to see what it is going to be of the same even the same collision how is going to how how one is going to see in s frame. And remember s prime frame is moving with a certain uniform velocity v with respect to the s frame. So, on top of that of course, on the back of my minds we we have the conservation of linear momentum in our in the collisional process. Well this is rather something very sacrosanct. I mean we are not going to violate all these conservation principles whether it is non-relativity or relativity or classical mechanics. I mean the conservation laws are conservation laws you are supposed to you are supposed to you are supposed to maintain them.  For example, if it is consider s prime frame you have m prime a and then the other mass b also of m prime. So, after collision the mass is going to be m prime plus m prime that is 2 m prime that is what I mean the total mass means conserved in the collision process in each of these frames. So, what does it mean here? Well from the linear momentum conservation what we have in this collisional process if you look at the s prime frame it is very simple it is going to be 0 is not it I mean it is they are moving that your identical masses the speeds are identical and opposite to each other moving parallel to the x axis x prime axis.  So, momentum so even if the if so that tells you that if you are going to have an inelastic collision in the end. So, you know the mass is going to be whatever it is and the velocity or the it is it has to be at rest in the s prime frame after the collision. Well something of that later on, but what about this the total final mass of the combination is final mass of the combination m prime plus m prime it is going to be 2 m prime there is nothing rocket science in that, but the only thing is that it is going to be maintained after collision that is what I want to say. So, that is what we have the thing. So, just one look one more slide to to recapitulate on this collisional process before the collision. So, we have the s prime frame where we have identical masses a and b they have similar masses they have the same mass m prime and they are moving opposite to each other parallel to the x prime axis and then one is observing the same thing from the s frame and then the s prime frame is moving with a certain velocity p with respect to the s frame. And from the s frame we are going to see the same collisional process, but then the velocities are not going to be different and then we have taken different masses here we do not know what we do not know whether the masses will be same or different, but let us see if from all these conservation laws what turns out to be what whether we have indeed different masses whether m 1 is separate is different from m prime. So, what happens after collision remember the collision is perfectly in elastic. So, what it means is a combined particle the combined particles will be at rest and they will stick together in the s prime frame. So, that is what they are they will stick together in the s prime frame the total mass is 2 m prime and they are at rest. Now, what will be the velocity with which these things will be seen to move from s frame? So, is it v we are going to find that out.  So, what about the velocity as we said. So, remember after collision well still after collision we are we are still considering s prime frame to be uniformly moving with a certain speed v with respect to the s frame. Now, in s prime frame a and b is at rest right is at rest. So, what does it tell you and then the s prime frame is moving with velocity v with respect to s. So, what is the velocity with which a and b will be seen to be moving from s frame well it is v that is what we have. So, in s frame we have these two particles they are sticking together since it is sticking together in s prime frame it is sticking together it is going to have mass m 1 plus m 2 and it is going to move with a certain velocity v which is same as the relative velocity between s and the s prime frame. So, just to recapitulate the entire collision process what you see in the top of this slide is the situation before the collision both in s prime frame and the s frame just have a nice look on the situation what is happening in the s prime frame and the situation that we have in the s frame that is the situation before collision and then after collision it is perfectly inelastic collision. So, it is sticking together in s prime frame and total mass is 2 m prime and it is at rest and then from s frame we are going to have the total mass to be m 1 plus m 2 the total mass is conserved here and it is moving with velocity v. So, next what we are going to do is that we are going to invoke the conservation of momentum in the s frame and remember if you invoke the conservation of momentum in the s prime frame it is going to be 0 is not it I mean the total momentum total linear momentum in the s prime frame is 0 and what is it if you invoke that same conservation of momentum in the s frame what is it going to be it is going to be m 1 u 1 plus m 2 u 2 that is the total momentum in what in the s frame before the collision and then after the collision it is going to be m 1 plus m 2 whole times v that is the velocity between this combination if a plus b or this m 1 plus m 2 is moving.  And then what you do is that you substitute for u 1 and u 2 the expressions from Einstein's velocity addition formula. So, you know that very well.  So, you are going to substitute these two velocities u 1 and u 2 in the expression given in the middle of your screen then given by this m 1 by m 2. Well, I am doing a little bit of the algebra a bit rather little bit of the algebra which only but if you miss this slide as a next we are not going to miss most of the physics. I am just doing it so that you will have the expression you can derive all these things by yourselves. So, what I have done is that now what will be m 1 plus m 2 in terms of all the quantities in the prime frame. So, it is 1 plus u prime v by c square divided by 1 minus u prime v by c square again what we do is that. So, we have derived up to the expression at the top of this slide again what we see is that a little more simplification actually you can derive this if you invoke again the Einstein's velocity addition formula. 1 minus u 1 square divided by c square and what is 1 minus u 2 square divided by c square. So, you put in what is u 1 and u 2 in terms of u prime v's and all these things you see rather very interesting expressions you can you can you can actually check that out. It is 1 minus u prime v by c square whole square divided by 1 plus u prime v by c square whole square. Now, from the top of the slide you can see that this is nothing but m 2 by m 1 whole square. Now, check what we have check the 1 on the left hand side. So, it is 1 minus u 1 square by c square divided by 1 minus u 2 square by c square and then on the extreme right hand side you have this m 2 by m 1 whole of square. This is very interesting these are the two expressions of the left hand side and the right hand side disregard the one in the middle. What do you see?  What do you again see? What you see is m 1 into 1 minus u 1 square by c square and you take a square root of that is equal to m 2 into 1 minus u 2 square by c square take a square root of that and that is it varies. That is not changing. So, what you see is that what you have here? So, m 1 and m 2 are the masses of identical particles. When the velocities are u 1 and u 2 respectively I mean when I say m 1 and m 2 are masses of identical particles and you know does not carry much of a meaning. I mean then you could say that identical particles how come they have different masses. But you see that they are moving at different speeds. So, this combination some rather interesting combinations of m 1 and u 1 of these speeds with which they are moving and then use that combination given by m 1 times 1 minus u 1 square by c square that square root of that that is a thing that is going to be invariant. Change the velocity the expression for the mass changes that is becoming invariant. So, what is invariant is this combination in whatever you know whatever mass and then it is corresponding velocity that is possible here. So, what we have? So, in a frame if v is the velocity of the particle the v is the velocity of the particle and m is its mass then the quantity that you will have m into 1 minus v square by c square root over of that that is going to be invariant. Now, this is very interesting why now what is this well rather what should be the invariant quantity let us check that out. Now, if you take the velocity with which the mass is moving to be 0. So, it is basically the mass is at rest. So, you consider you consider a frame in which the particle is at rest and then you measure the mass there. Now, if the mass is m 0 there. So, you plug in this quantities into the formula. So, m into 1 minus v square by c square. So, the mass is m 0 and the velocity is 0. See the 1 in the bracket 1 minus 0 square by c square and the root of that. So, that is going to be 1 no problem and then on the right hand side you have m 0. So, that is going to be the invariant quantity and m 0 is a mass of the same particle in a frame in which it is at rest. So, that is the formula we are looking for. See that if you know that the invariant quantity is actually m 0 here you can now relate the mass of a body which is moving with a certain velocity v as m is equal to m 0 which is the mass of the body in a frame in which is at rest and then divided by root over of 1 minus v square by c square. So, what we have is the mathematical formula and then this is very interesting of what the mass of the body will be if it is moving with a certain velocity v. Now, let us see how this is related. So, let us have a graphical illustration of this variation of mass and velocity. It is if you look at the graph on your screens what I have plotted here on the y axis is the mass and on the x axis I have v by c. It is basically it is the velocity in the units of velocity of light. So, it is someone it is starting from 0 to the speed of light. So, when v by c is 1 it is the speed of light. So, when v by c is 0 then you know that the body is at rest. Now, if we consider a mass of which is at rest which is m 0 to be of let us say 10 kg. Now, that is the green line. So, that is the green line and so, whether if you increase the velocity or not. So, that is m 0 it is the it is the rest mass. So, that is 10 kg, but as you increase the velocity see the red see the red curve see that at small speeds when the velocity of when the velocity of the body is not too high or it is too high not too high compared to the speed of light you see that it is it is increasing it is increasing and then all and then when it when it reaches towards the speed of light you see all of a sudden it increases very fast I mean it is almost explodes. So, you do see that the mass of the body is increasing in a certain way when when when the velocity is increased. Now, I also wish to remind you of one more thing it is that when you are at very small speeds when on basically when you are non-relativistic speeds when v is much much less than c you can immediately figure out look at the formula. For example, let us first look at the formula. So, m is equal to m 0 divided by root over of 1 minus v square by c square. So, when is much much when v is much much lesser than c v by c almost tends to 0. So, the denominator in this fraction is going to be 1. So, m will roughly be equal to m 0 when the velocity is much much less than the velocity of light and that is exactly what you see in the graph. So, when velocity is much much less than the velocity of light you are in the non-relativistic limit in the classical limit the classical world that we all live in you see that the red line actually converges with the green line. So, it actually fits with our experience. So, now let us see so so let us let us see what have another example of what the consequences of this of this formula would be. Let us just consider the rest mass of an electron the rest mass of an electron if we look up the tables it is something like 9.11 into 10 per minus 31 kilograms. Now, suppose this is moving with it is a pretty high velocity it is 80 percent the speed of light. So, will its mass be equal to you know how much will it mass what will it mass be will it be very near to 9.11 and 10 per minus 31 kilograms let us see. So, that is the rest mass and the velocity V here is 80 percent of the speed of light. So, what you see here will be m so that is the measured mass will be equal to m 0 divided by root over of 1 minus V square by C square. Now, m 0 you know what that is 9.11 into 10 per minus 31 kilograms and then 1 minus 0.8 square C square divided by C square. Now, this turns out to be something like 15 into 10 per minus 31 kilograms. So, you do see an increase in mass when you when the velocity is increasing and that too when the velocity is much much well it is actually 80 percent the speed of light it is quite a huge amount of which is quite a large velocity. So, by hope today I have you know you be able to cover some part of the consequences of special relativity we talked of Doppler effect in light and then the interesting phenomenon of the mass of a body varying with its velocity. In the next section what we will talk of is the relation between mass and energy and I will heavily draw upon what we have done so far about the consequences of relativity especially about the variation of mass with its velocity. Well, thank you very much.