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 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

Answer 1

The final speed of the particle after the piston is moved inward is

`vf=√(2v_0/v_0^2/v^2)`.

As the heavy piston is moved inward at a very low speed V, the small particle inside the hollow tube undergoes elastic collisions at both ends. Initially, when the piston is at a distance
`L_0`​ from the closed end, the particle has a speed
`v_0`. The kinetic theory of gases and the concept of elastic collisions can be applied to analyze this scenario.

When the piston is moved inward, it reduces the length of the tube, causing the particle to undergo more collisions with the moving piston. The final speed of the particle is determined by the interplay of its initial speed and the speed of the inward-moving piston (V).

The expression
`vf=√(2v_0/1+v_0^2/v^2)`represents the final speed of the particle, taking into account the elastic collisions and the changing dynamics within the tube as the piston is displaced. This equation reflects the conservation of kinetic energy during the collisions and provides a quantitative relationship for the final speed of the particle under the given conditions.