A pair of objects with masses 3m and m are connected by a hook and have a massless compressed spring between them. Initially the objects are at rest. Then the hook unlatches, the spring expands to its released length and the larger mass is then found moving left with speed v. which of the following three quantities is conserved in this process: total momentum p, total kinetic energy KE, total mechanical energy?

Question

A pair of objects with masses 3m and m are connected by a hook and have a massless compressed spring between them. Initially the objects are at rest. Then the hook unlatches, the spring expands to its released length and the larger mass is then found moving left with speed v. which of the following three quantities is conserved in this process: total momentum p, total kinetic energy KE, total mechanical energy?

Then, what is the final speed, V, of the smaller mass m?

Answer 1

momentum is conserved

the smaller mass is 1/3 of the larger
... so it must have 3 times the speed

Answer 2

To determine which quantities are conserved in this process, we need to consider the forces and energies involved.

1. Total momentum (p): Momentum is conserved when there is no external force acting on the system. In this scenario, the only forces involved are the internal forces between the masses and the spring. Therefore, total momentum is conserved.

2. Total kinetic energy (KE): Kinetic energy is generally not conserved unless the forces involved are conservative. In this case, the forces acting on the masses are non-conservative (spring force, hook force, and frictional forces in the environment). Therefore, total kinetic energy is not conserved.

3. Total mechanical energy: Mechanical energy, which includes both potential and kinetic energy, is conserved only in the absence of non-conservative forces. Since there are non-conservative forces acting on the masses, such as the spring force, hook force, and frictional forces, total mechanical energy is not conserved.

Now let's calculate the final speed of the smaller mass m (V):

In this process, the larger mass 3m moves left with speed v. Since momentum is conserved, the total momentum before and after the process will be equal.

Initially, the system is at rest, so the initial total momentum is zero. After the process, the larger mass 3m moves left with speed v, which means its momentum is (3m)(-v) = -3mv.

Since momentum is conserved, the smaller mass m must move right with an equal magnitude of momentum, but in the opposite direction. Therefore, the momentum of the smaller mass is (m)V, as it moves right.

Setting the initial total momentum to be equal to the final total momentum:

0 = -3mv + (m)V

Rearranging the equation, we get:

3mv = (m)V

Simplifying, we find:

V = 3v

Thus, the final speed of the smaller mass m is three times the speed v of the larger mass 3m.