Answer:
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
Let's consider the initial momentum of the system, which consists of the wagon, rider, and rock. The momentum is calculated by multiplying the mass of an object by its velocity. In this case, the total initial momentum is:
Initial momentum = (mass of the wagon + mass of the rider + mass of the rock) * velocity of the wagon
Initial momentum = (100 kg + 0.331 kg) * 0.45 m/s
Now, let's consider the final momentum of the system after the rock is thrown. There are two scenarios to consider:
(a) If the rock is thrown directly forward, the momentum of the rock in the forward direction is equal to its mass multiplied by its velocity. Since there are no external forces, the wagon's momentum remains unchanged.
Final momentum = (mass of the wagon + mass of the rider) * velocity of the wagon + (mass of the rock) * velocity of the rock (forward)
Final momentum = (100 kg) * v + (0.331 kg) * 154 m/s
(b) If the rock is thrown directly backward, the momentum of the rock in the backward direction is equal to its mass multiplied by its velocity. This time, the wagon's momentum will be affected in the opposite direction.
Final momentum = (mass of the wagon + mass of the rider) * velocity of the wagon + (mass of the rock) * velocity of the rock (backward)
Final momentum = (100 kg) * v - (0.331 kg) * 154 m/s
Using the conservation of momentum principle, the initial momentum is equal to the final momentum:
(100 kg + 0.331 kg) * 0.45 m/s = (100 kg) * v + (0.331 kg) * 154 m/s
Solving this equation for the velocity of the wagon (v) will give us the speed of the wagon after the rock is thrown in each scenario.
For case (a), where the rock is thrown directly forward, we solve the equation:
(100 kg + 0.331 kg) * 0.45 m/s = (100 kg) * v + (0.331 kg) * 154 m/s
Simplifying and solving for v, we find:
(100.331 kg) * 0.45 m/s - (0.331 kg) * 154 m/s = (100 kg) * v
v ≈ 0.404 m/s
Therefore, the speed of the wagon after the rock is thrown directly forward is approximately 0.404 m/s.
For case (b), where the rock is thrown directly backward, we solve the equation:
(100 kg + 0.331 kg) * 0.45 m/s = (100 kg) * v - (0.331 kg) * 154 m/s
Simplifying and solving for v, we find:
(100.331 kg) * 0.45 m/s + (0.331 kg) * 154 m/s = (100 kg) * v
v ≈ 0.428 m/s
Therefore, the speed of the wagon after the rock is thrown directly backward is approximately 0.428 m/s.