Aman holding a rock sits on a sled that is sliding across a frozen lake (negligible friction) with a speed of 0.480 m/s. the total mass of the sled, man, and rock is 96.0 kg. the mass of the rock is 0.310 kg and the man can throw it with a speed of 14.5 m/s. both speeds are relative to the ground. determine the speed of the sled if the man throws the rock forward (i. e. in the direction the sled is moving).
also determine the speed of the sled if he throws the rock directly backwards.

This is a problem of conservation of momentum

Momentum before throwing the rock: m*V = 96.0 kg * 0.480 m/s = 46.08 N*s

A) man throws the rock forward

=>

rock:
m1 = 0.310 kg
V1 = 14.5 m/s, in the same direction of the sled with the man

sled and man:
m2 = 96 kg - 0.310 kg = 95.69 kg
v2 = ?

Conservation of momentum:
momentum before throw = momentum after throw

46.08N*s = 0.310kg*14.5m/s + 95.69kg*v2

=> v2 = [46.08 N*s - 0.310*14.5N*s ] / 95.69 kg = 0.434 m/s

B) man throws the rock backward

this changes the sign of the velocity, v2 = -14.5 m/s

 46.08N*s = - 0.310kg*14.5m/s + 95.69kg*v2

v2 = [46.08 N*s + 0.310*14.5 N*s] / 95.69 k = 0.529 m/s