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The sneeze jet

Well, I'm back in at work after a lovely Christmas break. Lots of sunshine (we dodged the bad weather by going southwards for Christmas and then back north for New Year), beaches, playing with baby, and hacking back the jungle that sprung in the back garden in our 10 days' absence.

Benjamin has now pretty-well mastered sitting upright, which makes playtime a bit easier. Sometimes he leans too far and falls over, usually as a result of over-reaching for something that's caught his eye. A couple of days ago, he toppled backwards after a sneeze. Though it would be fun to think that the recoil from the momentum of the sneeze is what sent him over backwards, I think it is more likely that it just took him by surprise and he lost his balance.

A quick estimate shows the situation. A sneeze, in physics terms, is a jet of air. We can think of it in terms of conservation of momentum. A jet engine on an aircraft sends a high speed stream of gas backwards, and at the same time the aircraft exhibits a forward force - the two are related - the force is equal to the rate at which momentum is transferred to the stream of gas. To work this out we need to think what mass is moved backwards every second, and what is its velocity; the product of these gives the momentum transfer every second.

If we consider the jet as a cylinder whose ends are of area A (real jets aren't cylindrical, but let's not worry about that for a quick estimate), then in one second a length v metres of gas is emitted, where v is the velocity in metres per second. So the volume is Av, and the mass is pAv, where p is the density of the gas (for air it is conveniently about 1 kg per metre cubed). Since this gas moves at a velocity v, then the momentum per second is given by pAv times v, that is p A v squared.

So what force does Benjamin's sneeze provide? We have p=1 kg/m3 for air. Let's assume an area of about 2 cm2 for his open mouth (2 centimetres squared = 0.0002 metres squared). The velocity of a sneeze is pretty fast - a bit of googling and I find around 100 miles an hour (160 km/h or 44 m/s). For round numbers let's say 40 m/s. That means the force a sneeze gives him is 1 kg/m3 times 0.0002 m2 times 40 m/s squared which is about 0.3 newtons.

What does 0.3 newtons mean? That the force that gravity would exert on 30 grammes. Since Benjamin's legs weigh considerably more than this, the sneeze recoil isn't anywhere close to being sufficient to lift his legs off the ground and send him toppling backwards from a stable sitting position. Just maybe he was at the point of tipping and the sneeze is what sent him 'over the edge', but that would be quite a coincidence. So, while it was fun to think a sneeze can catapult you backwards, it's not going to happen outside of Roadrunner cartoons and that ilk.




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