Question:

A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from the closed end is L=L0, the particle speed is v=v0. The piston is moved inward at a very low speed V such that V<<dL/v0, where dL is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from L0 to 1/2L0

After each collision with the piston, the particle speed increases by 2V.

If the piston moves inward by dL, the particle speed increases by 2V(dL/L)

The rate at which the particle strikes the piston is v/L

Solution:

Correct option is C. The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from L0 to 1/2L0.
Rate at which the particle strike the piston = f = v/2x
If x = L then f = v/2L
Rate of change of speed of particle = dv/dt = f × 2V
dv = v/2x 2V dt
dv = v/x (-dx) (because piston is moving inward)
∫v0v dv/v = ∫L0x -dx/x
ln(v/v0) = -ln(x)
ln(v/v0) = ln(1/x)
v = v0/x
When x = L0/2, then v = v0(L0/L0/2) = 2v0
KE at x = L0/2 = Kf = 1/2 × m × 4v0^2
KE at x = L0 = Ki = 1/2mv0^2
∴ kf/ki = 4