Conservation of Momentum and Energy in Elastic Collision Demonstrations

Conservation of Momentum and Energy in Elastic Collision Demonstrations Michael B. Ottinger Missouri Western State University April 26, 2014

Newton’s Cradle Demonstration • Five or more balls of equal size and mass suspended in a line. • Pull one back and release  one exits the far side • Pull two back and release  two exit the far side Why does this work? Answer from Textbook: “This is the only way to Conserve both Momentum and Kinetic Energy.”

“This is the only way to Conserve both Momentum and Kinetic Energy.” • Answer is misleading • What would happen if the first ball had twice the mass of the other balls? • Would two balls exit the far side with the same initial speed? (This also conserves both momentum and kinetic energy)

Analysis of Motion: One Ball • Series of Elastic Collisions • Conservation of momentum • Conservation of Kinetic Energy • Equal Masses

Conservation of Momentum: Conservation of Kinetic Energy: Final Speeds:

With multiple collisions momentum and energy are transferred between each ball. • Initially ball on the right has the momentum and ball on left is at rest • After collision ball on right is at rest and ball on left has momentum. Momentum is transferred through each ball until last ball has no collision and moves away.

Analysis of Motion: Two Balls • Series of Elastic Collisions • Conservation of momentum • Conservation of Kinetic Energy • Equal Masses

Analysis of Motion: Two Balls • Two balls move at same initial speed. • No force between balls. • Multiple elastic collisions 2 to 3, 3 to 4, 4 to 5  5 moves out with speed v 1 to 2, 2 to 3, 3 to 4  4 moves out with speed v

Analysis of Motion: Unequal Mass • First Ball has mass M=2m • Other Balls have mass m • Series of Elastic Collisions • Conservation of momentum • Conservation of Kinetic Energy ?

Analysis of Motion: Unequal Mass Consider the First Collision: Conservation of Momentum Conservation of Kinetic Energy Final Speeds:

Analysis of Motion: Unequal Mass • Energy/Momentum transferred to second ball will transfer from it to all successive balls. Final ball exits with speed 4v1/3. • First Ball is not at rest, so it will collide again with Second Ball at rest.

Analysis of Motion: Unequal Mass Conservation of Momentum (2nd collision): Conservation of Kinetic Energy Final Speeds: Energy/Momentum transferred through each ball and penultimate leaves at v=4/9v1

Analysis of Motion: Unequal Mass First ball still not at rest, so it collides again, and again. Results: To conserve momentum and energy, each ball will exit at successively smaller speeds:

Conclusion: Newton’s Cradle • When the balls are of equal mass, momentum and energy are conserved through successive collisions. In these collisions all of the momentum/energy of the moving ball is transferred to a stationary ball. • The same number of balls exit as were released not “because this is the only way to conserve momentum and energy” but because of the successive collisions. • Balls of unequal mass will not transfer all of the momentum/energy so each ball travels with a finite final speed.

A Few Other Interesting Scenarios • What if the final ball has twice the mass? • What if an interior has twice the mass? • What if the first ball has half of the mass? • What if the last ball has half of the mass? Newton’s Cradle can produce many opportunities for students to analyze the conservation of momentum and energy in elastic collisions.

A Second Interesting Demonstration Drop two balls from a given height with the small ball on top of the larger ball. The small ball will bounce back to a much higher point than it was dropped.

Analysis of Motion • Balls fall with the same acceleration and velocity before hitting the ground. • Pre-collision velocity calculated from conservation of energy or kinematic equations. • Consider two elastic collisions: First: Lower Ball with Ground Second: Upper and Lower Balls

Analysis of Motion First Collision: Since Ground is much more massive than the ball, the speed of the ball will remain the same, but change direction.

Analysis of Motion Second Collision: Balls are now moving at he same speed, but in opposite directions. Conservation of Momentum: Conservation of Kinetic Energy: Final Speeds:

Analysis of Motion Final Height of Second Ball is found from conservation of energy: Example: Basketball (M=0.623 kg) Tennis Ball (m=0.043 kg)

Questions?

Conservation of Momentum and Energy in Elastic Collision Demonstrations