A wagon is coasting at a speed vA along a straight and level road. When twenty-five percent of the wagon's mass is thrown off the wagon, parallel to the ground and in the forward direction, the wagon is brought to a halt. If the direction in which this mass is thrown is exactly reversed, but the speed of this mass relative to the wagon remains the same, the wagon accelerates to a new speed vB. Calculate the ratio vB/vA.
To solve this problem, we need to apply the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant if no external forces are acting on it.
Let's denote the mass of the wagon as M and the initial speed of the wagon as vA. After throwing off 25% of its mass, the remaining mass of the wagon is 0.75M.
Now, we can analyze the situation step by step:
1. When 25% of the wagon's mass is thrown off forward, the wagon experiences a change in momentum in the opposite direction. Let's denote the mass of the thrown-off part as m and the speed at which it is thrown off as v.
The initial momentum of the wagon is M * vA.
The initial momentum of the thrown-off mass is m * v.
The final momentum of the wagon after the throw is (0.75M) * vA since 25% of the wagon's mass is thrown off. The final momentum of the thrown-off mass is 0 since it comes to a halt.
According to the conservation of momentum, the total initial momentum should be equal to the total final momentum:
M * vA + m * v = (0.75M) * vA + 0
Rearranging the equation, we get:
M * vA + m * v = (0.75M) * vA
2. Now, let's consider the case when the thrown-off mass is reversed but the speed relative to the wagon remains the same. In this case, the mass m is still thrown off forward but with the same speed v, and thus the change in momentum is in the same direction as before.
The initial momentum of the wagon is (0.75M) * vA (since it has lost 25% of its mass).
The initial momentum of the thrown-off mass is m * v.
The final momentum of the wagon is M * vB, as given in the problem.
According to the conservation of momentum again, we have:
(0.75M) * vA + m * v = M * vB
3. We can now solve these two equations simultaneously to find the ratio vB/vA:
M * vA + m * v = (0.75M) * vA --> Equation 1
(0.75M) * vA + m * v = M * vB --> Equation 2
From Equation 1, we can solve for m * v:
m * v = (0.75M) * vA - M * vA
m * v = M * vA * (0.75 - 1)
m * v = -0.25 * M * vA
Substituting this value in Equation 2:
(0.75M) * vA - M * vA - 0.25 * M * vA = M * vB
(0.5M) * vA = M * vB
vB/vA = 0.5
Therefore, the ratio vB/vA is 0.5 or 1/2.