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## 08 December 2016

### Momentum and kinetic energy when people push off each other

A mother and her son are initially at rest next to each other on an ice rink on which friction is negligible.  The mother’s mass is twice the son’s mass.  They push off of each other, causing them to glide apart.

1. Is the magnitude of the two skaters' total momentum larger before or after the push?

Simplest answer: Momentum is conserved in a collision.  Total momentum is zero before the push because nothing moves.  So, afterward the total momentum must likewise be zero.

Deeper answer: How do we know momentum is conserved here?  Because no net external force acts.  The normal force and weight cancel because there is no vertical speed change.  The only horizontal forces are the forces of the skaters on each other -- these are internal to the two-skater system.

Corkscrew-thinking too-deep answer that is nonetheless fully correct:  J=Ft.  The forces of each skater on the other are equal due to Newton's third law.  The time of collision is the same for both skaters -- otherwise we wouldn't be in the same collision.  So impulse is the same for both.  Impulse is the change in momentum, meaning both skaters have the same amount of momentum after the push.  Since they move in opposite directions, the vector sum of the objects' momentums is zero.

Not fully correct: Since the skaters move in opposite directions with equal momentums, the total momentums subtract to zero.  (How do we know that the skaters have equal momentums after the push?  That requires the corkscrew thinking above.  Without reference to N3L and the impulse-momentum theorem, there's no evidence that the have equal and opposite momentums.)

2. Is the total kinetic energy of the two skaters larger before or after the push?

Simplest answer: Total kinetic energy is zero before the collision because there's no motion at all.  After the collision, both skaters move, so both have kinetic energy.  The system kinetic energy is the (scalar) sum of each skater's kinetic energy, which is not zero.  So larger KE after the push.

Deeper answer: Why is mechanical energy not conserved here?  After all, as in question 1, we can show that no net external force even acts, let alone does work, on the skaters.  Well, it's the internal energy from the skaters' muscles that is converted to kinetic energy.  The skaters' bodies lower their internal energy by converting ATP to ADP, in the process causing their arms to apply a force through a distance on the other skater.  That's mechanical work, which changes each skater's kinetic energy.