Conservation of Linear Momentum and Newton's Third Law

Problem

Demonstrate how Newton’s Third Law and the Conservation of Momentum affect movement.

Materials

• Two skateboards
• Medicine ball

Procedure

1. Have two people sit on the skateboards a few feet apart, facing each other and with their feet off the ground.
2. Have one person throw the ball to the other. How does the person who threw the ball move after the ball is thrown? What about the person who catches the ball?

Results

The person who throws the ball will roll backwards. When the second person catches the ball, she will move in the direction the ball was going.

Why?

There are two principles at work here: Newton’s Third Law of Motion and the conservation of momentum.

Newton’s Third Law states that for every action, there is an equal and opposite reaction. If you push against a wall, the wall pushes back against you with the same amount of force. When you’re sitting on a skateboard and you throw a ball, you move in the opposite direction. This, by the way, is exactly how rockets work. They are designed to keep throwing stuff—ignited propellant, in this case—out one end. The rocket responds by moving in the opposite direction. If you loaded your skateboard up with dozens of balls and kept throwing them, one after the other, you could propel yourself some distance across the floor. Running out of balls would be the same as running out of fuel!

Another way to look at it is to keep in mind the fact that momentum must always be conserved. Momentum is a property of moving objects that comes from multiplying together the mass and velocity, mv. When we say momentum must be conserved, what we mean is that a system’s momentum at one time must equal the total momentum at a later time. On the skateboard, your momentum is initially zero because you aren’t moving. When you throw the ball, it achieves a momentum of m_bv_b (where m_b is the ball’s mass and v_b is its velocity). To keep everything balanced, your momentum must exactly balance the ball’s to keep the total momentum at zero. Writing that as an equation, you get:

m_bv_b = —m_yv_y

where m_y is your mass and v_y is your velocity. The negative sign means you have to move in the opposite direction as the ball.

When your friend catches the ball, something similar happens. The ball comes in with a certain amount of momentum, m_bv_b. When she catches it, some of that momentum gets transferred to her and she is sent rolling backward:

m_bv_b = (m_f + m_b )v_f

where v_f is the final velocity of the combined mass (m_f + m_b) of your friend and the ball. Notice there’s no negative sign this time: your friend moves in the same direction as the ball.

Going Further

Repeat the above experiment with balls of different weights. Keep track of how far both you and your friend move each time. What do you notice? How would you explain your observations using the principle of conservation of momentum?