A $2\mathrm{.}0-$g particle moving at $8\mathrm{.}2$ m$/$s makes a perfectly elastic head$-$on collision with a resting $1\mathrm{.}0-$g object.

$($a$)$ Find the speed of each particle after the collision.

$2\mathrm{.}0$ g particle$$

Apply momentum conservation and assume a head$-$on elastic collision to find the final velocities from the initial velocities. m$/$s$1\mathrm{.}0$ g particle$$

Apply momentum conservation and assume a head$-$on elastic collision to find the final velocities from the initial velocities. m$/$s

$($b$)$ Find the speed of each particle after the collision if the stationary particle has a mass of $10$ g$.$

$2\mathrm{.}0$ g particle$$

Apply momentum conservation and assume a head$-$on elastic collision to find the final velocities from the initial velocities. m$/$s$10\mathrm{.}0$ g particle$$

Apply momentum conservation and assume a head$-$on elastic collision to find the final velocities from the initial velocities. m$/$s

$($c$)$ Find the final kinetic energy of the incident $2\mathrm{.}0-$g particle in the situations described in parts $($a$)$ and $($b$).$

KE in part $($a$)$

Use the final velocity you calculated for the $2\mathrm{.}0-$g particle in part $($a$)$ to find its final kinetic energy. JKE in part $($b$)$