A head-on, elastic collision between two particles with equal initial speed v leaves the more massive particle (mass m1) at rest. find the ratio of the particle masses

1/3 The key thing to remember about an elastic collision is that it preserves both momentum and kinetic energy. For this problem I will assume the more massive particle has a mass of 1 and that the initial velocities are 1 and - 1. The ratio of the masses will be represented by the less massive particle and will have the value "r" The equation for kinetic energy is E = 1/2MV^2. So the energy for the system prior to collision is 0.5r (-1) ^2 + 0.5 (1) ^2 = 0.5r + 0.5 The energy after the collision is 0.5rv^2 Setting the two equations equal to each other 0.5r + 0.5 = 0.5rv^2 r + 1 = rv^2 (r + 1) / r = v^2 sqrt ((r + 1) / r) = v The momentum prior to collision is - 1r + 1 Momentum after collision is rv Setting the equations equal to each other rv = - 1r + 1 rv + 1r = 1 r (v+1) = 1 Now we have 2 equations with 2 unknowns. sqrt ((r + 1) / r) = v r (v+1) = 1 Substitute the value v in the 2nd equation with sqrt ((r+1) / r) and solve for r. r (sqrt ((r + 1) / r) + 1) = 1 r*sqrt ((r + 1) / r) + r = 1 r*sqrt (1+1/r) + r = 1 r*sqrt (1+1/r) = 1 - r r^2 * (1+1/r) = 1 - 2r + r^2 r^2 + r = 1 - 2r + r^2 r = 1 - 2r 3r = 1 r = 1/3 So the less massive particle is 1/3 the mass of the more massive particle.