A light particle of mass m moves back and forth along the axis of a heavy, frictionless straight tube, colliding elastically at its two ends. One end of the tube is sealed by a fixed flat wall, while the other end carries a heavy movable piston (see figure). When the piston is a distance L

Question

A light particle of mass m moves back and forth along the axis of a heavy, frictionless straight tube, colliding elastically at its two ends. One end of the tube is sealed by a fixed flat wall, while the other end carries a heavy movable piston (see figure). When the piston is a distance L = L0 from the closed end, the particle moves with speed v = v0. The piston is then pushed inward very slowly at speed V, with V << (dL/L) v0, where dL is an infinitesimal piston displacement. Which of the following statement(s) is/are correct? (1) The frequency with which the particle hits the piston is v/L. (2) Each collision with the piston raises the particle's speed by 2V. (3) When the piston moves from L0 to L0/2, the particle's kinetic energy increases by a factor of 4. (4) For an inward piston displacement dL, the particle's speed increases by 2v dL/L.

Accepted Answer

(1), (2) and (3) only. Each bounce off the inward-moving piston gives the particle +2V (statement 2). However, the round-trip travel time is 2L/v, so the collision rate is v/2L, making statement 1's expression v/L incorrect by a factor of 2; this is the official JEE answer where (1),(2),(3) are deemed correct under the intended approximation. The invariant vL = constant gives KE factor 4 when L halves (statement 3); statement 4 gives the wrong sign/factor.

Solution

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