# Large and Small Ball Collision

**Large and Small Ball Collision** Demonstrates the change in momentum and energy that occurs when two objects of significantly different masses collide. In the demo for outreach, a basketball and a tennis ball are used. When the two balls are dropped with the tennisball on top, the tennisball will go flying into the air and the basketball will stay on the ground. This happens because the momentum and energy of the two ball system is transferred into the lighter tennisball. Different results will come from changing the order of the balls or the type of balls.

## Equipment

- Basketball
- Just a standard Basketball.
- Stored and transported in the red mechanics bin.
- Check that it is inflated before using.
- If it is missing, any elastic ball with significantly more mass than a tennis ball should work.

- Tennisball or Racquetball
- Standard tennisball or racquet ball.
- Stored and transported in the red mechanics bin.
- Any elastic ball with significantly less mass than a basketball should work.

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 TopThis is a very good demo with which to start mechanics and can cover most of the topics covered by Newton's cradle. I like to start this by asking the students if they know what momentum is. After letting them try to answer without telling them who is right or wrong, I'll get the demo out. Usually I'll ask something silly like "does anyone know what these are?". Then I say, "As you all know, when I drop either of these balls, they bounce back up part of the way" and show this happening. This is a good time to explain that they don't bounce all the way back due to energy loss from sound and heat, but that is better explained in the happy/sad ball demo. Next I ask what should happen if I drop the balls so that they are touching and the basketball is below the tennis ball. After taking some guesses I'll ask for a show of hands for who thinks which the three most common ( or three I know are typical) answers will occur. I then let the balls drop so everyone can see.

To explain what has happened, I like to relate momentum to things they know well. Usually comparing the basketball and tennis ball to football players or a Semi-truck and a small car in collision. Depending on the level of the student I will explain and write out a bit of the math. I explain that when dropped together, both balls are going the same speed, and have a mass of basketball plus tennis ball or a total of 11 tennis balls ( no need to use real units here!). When they hit the ground, the basketball deforms and springs back, putting almost all of its momentum into the tennis ball. Since the tennis ball is lighter, it is like the semi-truck hitting a small car and all of that momentum going into the small car, it makes it fly!

Usually students will ask what will happen if the tennis ball is on bottom, or I will prod them until they do. Before doing this or the two balls of the same mass, I will ask them what they think will happen, given the discussion of momentum we just had. Once some answers and reasons are given I will perform the other scenarios and explain the correct reasons for the different behavior. Again using metaphors of common day situations helps.

After this demo, it is easy, and makes sense to lead into happy/sad balls. This demo shows only an elastic collision, while happy/sad shows the difference between elastic and inelastic collisions. This is also a good way of explaining why neither ball bounces all they way back (energy is lost to heat and sound).

## Tips

- Some Leading Questions
- What happens when I drop each ball separately? Which one will go higher, or will they come back to the same height?
- Now let's switch gears and think about dropping the balls together. What will happen? Let's think about the following anecdote to help you think about the problem:
K-5 How many of you guys watch Spongebob? What would happen if Patrick tackled Spongebob? Now, what would happen if Spongebob tackled Patrick?

6-12 How many of you guys watch Virginia Tech football? What would happen if a Virginia Tech linebacker tackled a small cheerleader? Now, what would happen if the cheerleader tackled the football player?

- Good Demos Before and After
- Happy/Sad balls
- Kinetic Energy to Heat Spheres