Question

A projectile with a mass M and a velocity v(t = 0) = 0 explodes into two pieces. The energy gain from the explosion is E. What is the ideal mass distribution so that for one of the particles

a) The kinetic energy is maximised.
b) The velocity is maximised.
c) The momentum is maximised.

Hint: Use what you learned about the center of mass system.

Answer 1

(a) The kinetic energies are K₁=p^2/2m₁, K₂=p^2/2m₂ so they are in the ratio K₁/K₂=m₂/m₁. To get K₁≫K₂ we need m₁≪m₂, which is the same requirement as for maximising velocity.

(b) The velocity of one part is maximum when its mass is much less than that of the other. Again this follows from conservation of momentum : m₁v₁=m₂v₂ therefore v₁≫v₂ if m₁≪m₂.

(c) The momenta will be equal (p₁=p₂=p) regardless of mass distribution. This follows from conservation of momentum.