Large and Small Ball Collision

This demo is used as a simple introduction to conservation of momentum, and can be used to discuss conservation of energy. When the tennis ball and basketball are dropped touching one another, the momentum of the system is:

$$ p_{total}=m_{tennis} v_{tennis} + m_{basket}v_{basket}$$

Since both are moving at the same velocity, this simplifies to

$$ (m_{tennis}+m_{basket})v $$

We know that the basketball has has about 10 times the mass of a tennis ball ^[1][2] so we can further simplify

$$ (m_{tennis} + 10m_{tennis})v \Rightarrow 11m_{tennis}\cdot v $$

When the basketball hits the ground, it deforms both where it contacts the ground and where the tennis ball is in contact with it. Since both balls are elastic, both rebound to their original shapes. When this happens, most of the momentum is transferred to the ball on top. Since the collision in this situation is elastic, momentum is conserved, meaning the momentum of both balls right before hitting the floor is equal to the momentum of both balls right after the collision. Mathematically...

$$ m_{initial} \cdot v_{initial} = m_{final} \cdot v_{final} $$

So we can set up the before- and after-collision equations to solve for the velocity of the tennis ball in the end as compared to the initial velocity. (I will be assuming the basketball ends up with a final velocity of zero. While this is not exactly the case, it is close enough for our purposes. By measuring the height to which the balls travel, one could find a more exact value).

$$ 11m_{tennis}\cdot v_{initial}=m_{tennis} \cdot v_{final} + 10m_{tennis} \cdot 0 $$

$$ {m_{tennis} \over m_{tennis}}11v_{initial}=v_{final} $$

Since the mass of the tennis ball cancels out, this shows that the final velocity is 11 times larger than the initial velocity! (Because the collisions are not perfectly elastic, the true velocity is less than this due to energy lost making sound and heat generated in deformation).

If the balls are reversed (tennisball under the basketball)

$$ ({1 \over 10}m_{basketball}+m_{basketball})\cdot v_{initial}={1 \over 10}m_{basketball}\cdot 0+m_{basketball}\cdot v_{final} $$

$$ {m_{basketball}\over m_{basketball}}{11 \over 10}\cdot v_{initial}= v_{final} $$

So in this scenario, the basketball only goes 10% faster! (This is actually small enough that it may be negated by the loss of energy due to sound and the increased heat made from the larger deformation of the tennis ball).

Putting two tennisballs or two basketballs together results in the simple equation

$$ {m_{ball}\over m_{ball}}{2}\cdot v_{initial}= v_{final} $$

Meaning that two balls of the same mass will launch one at about twice the initial speed if all energy loss is ignored.

Back to Top