Question

A mad engineer builds a cube,$2.5\mathrm{m}$on a side, in which $6.2-\mathrm{cm}$diameterrubber balls are constantly sent flying in random directions by vibrating walls. He will award a prize to anyone who can figure out how many balls are in the cube without entering it or taking out any of the balls. You decide to shoot $6.2-\mathrm{cm}$diameter plastic balls into the cube, through a small hole, to see how far they get before colliding with a rubber ball. After many shots, you find they travel an average distance of $1.8\mathrm{m}$. How many rubber balls do you think are in the cube?

Short answer

Number of balls are in the cube is$508$.

Step-by-step solution

Formula for free mean path

Mean free path is,

$\mathrm{\lambda }=\frac{1}{4\mathrm{\pi }\sqrt{2}\mathrm{N}/{\mathrm{Vr}}^2}=\frac{\mathrm{V}}{4\mathrm{\pi }\sqrt{2}{\mathrm{Nr}}^2}$

We can solve for the number of particles $\mathrm{N}$using this expression.

$\mathrm{N}=\frac{\mathrm{V}}{4\mathrm{\pi }\sqrt{2}{\mathrm{\lambda r}}^2}$

Calculation for number of balls in the cube

Cube of side $\mathrm{a}=2.5\mathrm{m}$,

The diameter$\mathrm{d}$of the ball, not the radius,

we can substitute these in the expression,

$\mathrm{N}=\frac{{\mathrm{a}}^3}{4\mathrm{\pi }\sqrt{2}\mathrm{\lambda }{\left(\frac{\mathrm{d}}{2}\right)}^2}=\frac{{\mathrm{a}}^3}{\mathrm{\pi }\sqrt{2}{\mathrm{\lambda d}}^2}$

$\mathrm{N}=\frac{{(2.5)}^3}{\mathrm{\pi }\sqrt{2}(1.8){(0.062)}^2}$

$\mathrm{N}=508$