A Laboratory Exercise in Indirect Measurement

Discussion: Modern physics depends heavily on indirectly determining physical properties of objects. The following activity may help convince students that indirect determinations are important methods of obtaining accurate information. This exercise can be used as an introduction to a discussion of the Rutherford model of the atom. This activity simulates an experiment in particle physics where a target material would be bombarded by high speed particles, and the collisions studied. It gives you a chance to use a "Monte Carlo" technique.

Problem: Indirectly determine the radius of a single target circle.

Use copies of the circle sets on the following pages. Place the circled paper on the floor, face down over a sheet of carbon paper. Working in pairs drop marbles or ball bearings from head height so that they hit the paper. The sphere must be caught after the first bounce. Repeat this at least 100 times. It may be more convenient to drop the marbles from just above the paper, however, one should then take care to distribute the hits as randomly as possible over the entire target area. It is "OK" to miss the paper from time to time. Those points will naturally be excluded form the data set. (Note the paper is 21.5 cm by 28.0 cm.)

Analysis: Count the total number of dots on the paper (total hits), as well as the number of dots just completely within a circle (circle hits). Determine the total area of the paper, and count the total number of circles on the paper. If the circles are of uniform size and the hits are randomly distributed, then one can assume:

(circle hits)/(hits) = (area of all circles)/(rectangular area)

Therefore, you can calculate the total area of all circles. From this you can calculate:

area of one circle = (area of all the circles)/(number of circles)

The area of one circle can be used to calculate the radius of a circle (area = pi*radius*radius). This calculated radius may then be compared with a direct radius measurement.


1. What is the radius of one circle?

2. How does the indirect measurement (using the marbles) compare to a direct measurement (such as using a ruler)?

Included in the Topics in Modern Physics, May 1990, and Catching the Sun, 1992, Fermilab.