We know that momentum and kinetic energy are conserved in elastic collisions.

Conservation of momentum:

$\overline{){{\mathbf{m}}}_{{\mathbf{a}}}{{\mathbf{v}}}_{\mathbf{a}\mathbf{0}}{\mathbf{+}}{{\mathbf{m}}}_{{\mathbf{b}}}{{\mathbf{v}}}_{\mathbf{b}\mathbf{0}}{\mathbf{=}}{{\mathbf{m}}}_{{\mathbf{a}}}{{\mathbf{v}}}_{\mathbf{a}\mathbf{f}}{\mathbf{+}}{{\mathbf{m}}}_{{\mathbf{b}}}{{\mathbf{v}}}_{\mathbf{b}\mathbf{f}}}$

Relative velocities after an elastic collision:

$\overline{){{\mathbf{v}}}_{\mathbf{a}\mathbf{0}}{\mathbf{-}}{{\mathbf{v}}}_{\mathbf{b}\mathbf{0}}{\mathbf{=}}{{\mathbf{v}}}_{\mathbf{b}\mathbf{f}}{\mathbf{-}}{{\mathbf{v}}}_{\mathbf{a}\mathbf{f}}}$

A 10.0-g marble slides to the left at a speed of 0.400 m/s on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0-g marble sliding to the right at a speed of 0.200 m/s.

(a) Find the velocity of each marble (magnitude and direction) after the collision. (Since the collision is head-on, all motion is along a line.)

(b) Calculate the change in momentum (the momentum after the collision minus the momentum before the collision) for each marble. Compare your values for each marble.

(c) Calculate the change in kinetic energy (the kinetic energy after the collision minus the kinetic energy before the collision) for each marble. Compare your values for each marble.

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