Question

A system of three masses, m₁ , m₂ and m₃ and a spring (of spring constant k) is set up, as shown, an a straight narrow and smooth track.

The mass, m₁ , starts moving with a veloicty u (m/s) and collides elastically with the mass m₂ ; A little later, when the spring has been compressed by an amount x₀ (metre), the masses m₂ and m₃ are moving with the same veloicty, say, v. The velocity v, and the spring constant, k, of the spring, are then given respectively by

(1) \(V =\frac{u}{3}ms^{},k=(\frac{1}{30}\frac{u^2_0}{x^2_0})N/M\)

(2) \(V =\frac{u}{3}ms^{-1},k=(\frac{1}{80}\frac{u^2_0}{x^2_0})N/M\)

(3) \(V =\frac{u}{2}ms^{-1},k=(\frac{1}{30}\frac{u^2_0}{x^2_0})N/M\)

(4) \(V =\frac{u}{3}ms^{-1},k=(\frac{1}{30}\frac{u^2_0}{x^2_0})N/M\)

Answer 1

(4) \(V =\frac{u}{3}ms^{-1},k=(\frac{1}{30}\frac{u^2_0}{x^2_0})N/M\)

The masses m₁ and m₂ being equal, their velocities, after m₁ collides elastically with m₂ , are zero and u, respectively. 

A little later, when m₂ and m₃ have the same veloicty, v, and the spring has been compressed by an amount x₀ , we have, by the laws of conservation of momentum and energy.