A man holding a rock sits on a sled that is sliding across a frozen lake (negligible friction) with a speed of 0.500 m/s. The total mass of the sled, man, and rock is 96.5 kg. The mass of the rock is 0.330 kg and the man can throw it with a speed of 16.5 m/s. Both speeds are relative to the ground. Determine the speed of the sled (in m/s) if the man throws the rock forward (i.e., in the direction the sled is moving).
To solve this problem, we can use the principle of conservation of momentum. Initially, the total momentum of the system (man, sled, and rock) is the momentum of the sled, man, and rock moving forward together.
Let the initial velocity of the sled, man, and rock be v, the final velocity of the sled after the man throws the rock forward be V, and the velocity of the rock thrown by the man be v_r.
The initial momentum of the system is:
(before) P_initial = (total mass)*(initial velocity) = 96.5 kg * 0.500 m/s = 48.25 kg·m/s
The final momentum of the system is:
(after) P_final = (mass of sled and man)*(final velocity) + (mass of rock)*(velocity of the rock)
P_final = 96.17 kg * V + 0.330 kg * 16.5 m/s
According to the conservation of momentum, the initial momentum must be equal to the final momentum:
P_initial = P_final
48.25 = 96.17*V + 0.330*16.5
After solving the equation, we find:
V ≈ 0.511 m/s
Therefore, the speed of the sled after the man throws the rock forward is approximately 0.511 m/s.