A mad engineer builds a cube, 2.5 m on a side, in which 6.2-cm-diameter rubber 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-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 m. How many rubber balls do you think are in the cube?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to estimate the number of rubber balls in the cube. This can be approached using the concept of mean free path, which is the average distance a particle travels before colliding with another particle. The mean free path formula is: $\lambda =\frac{1}{n\pi d^2}$, where $\lambda$ is the mean free path, $n$ is the number density of the rubber balls, and $d$ is the diameter of the balls.
2. Step 2: Rearrange the formula to solve for $n$, the number density of the rubber balls. The rearranged formula is: $n=\frac{1}{\lambda \pi d^2}$. Here, $\lambda$ is given as 1.8 m, and $d$ is the diameter of the balls, which is 6.2 cm or 0.062 m.
3. Step 3: Calculate the number density $n$ using the formula. Substitute the values: $n=\frac{1}{1.8\times \pi \times {0.062}^2}$. This will give the number of rubber balls per cubic meter.
4. Step 4: Calculate the total number of rubber balls in the cube. The volume of the cube is ${2.5}^3$ cubic meters. Multiply the number density $n$ by the volume of the cube to find the total number of rubber balls: $N=n\times {2.5}^3$.
5. Step 5: Interpret the result. The calculated value of $N$ represents the estimated number of rubber balls in the cube. This approach assumes that the plastic balls interact with the rubber balls in the same way as the rubber balls interact with each other, and that the mean free path is accurately measured.

Key Concepts

Volume of the Cube

The volume of a cube is calculated using the formula V = side³, where 'side' is the length of one edge. In this scenario, the cube has a side length of 2.5 m, which means its volume is 2.5³ = 15.625 m³. This volume is crucial for determining how many rubber balls can fit inside the cube based on their individual volumes.

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where 'r' is the radius. For the rubber balls with a diameter of 6.2 cm, the radius is 3.1 cm (0.031 m). Calculating the volume of one rubber ball allows us to estimate how many such balls can fit into the total volume of the cube.

Collision Probability and Average Distance

The average distance traveled by the plastic balls before colliding with a rubber ball provides insight into the density of the rubber balls within the cube. If the plastic balls travel an average of 1.8 m, this distance can be related to the number of rubber balls present, as a higher density of balls would likely result in more frequent collisions, thus affecting the average distance traveled.