Question

A mass (m) moves along the x-axis with velocity 2v. Another particle of mass (2m) moves with velocity v along the y-axis.

1) If the two objects collide inelastically and merge, how much kinetic energy is lost to heat?

2) If instead they collide elastically and it turns out that m still moves purely along the x-axis, what is it's velocity in the final state?

It may help to find the velocities of these particles in the center of mass frame before attempting to solve

Answer 1

(1) If the two objects collide inelastically & merge, then its kinetic energy lost to heat will be given as :

K.E_lost = K.E_initial - K.E_final

K.E_lost = (1/2) m₁ v₁² - (1/2) (m₁ + m₂) v₂²

K.E_lost = (1/2) m (2 v)² - (1/2) (3 m) v²

K.E_lost = (1/2) [4 m v² - 3 m v²]

K.E_lost = (1/2) m v²

(2) For an elastic collision, we have

v_1,f = [(m₁ - m₂) / (m₁ + m₂)] v_1,i + [(2 m₂) / (m₁ + m₂)] v_2,i

v_1,f = - (m / 3 m) (2 v) + (4 m / 3 m) v

v_1,f = - [(2/3) + (4/3)] v

v_1,f = (2 / 3) v

v_1,f = 0.66 v