A 5.00-kg ball, moving to the right at a velocity of + 2.00 m/s on a frictionless table, collides head-on with a stationary 7.50kg ball. find the final velocities of the ball if the collision is elastic

Using the law of conservation of momentum:

m₁u₁ + m₂u₂ = m₁v₁ + m₂ v₂; u₂ = 0

5 x 2 = 5v₁ + 7v₂

10 = 5v₁ + 7v₂

v₂ = (10 - 5v₁) / 7

Elastic collision so kinetic energy conserved:

1/2 x 5 x 2² = 1/2 (5v₁² + 7v₂²)

20 = 5v₁² + 7 ((10 - 5v₁) / 7) ²

140 = 35v₁² + 5v₁² + 100 - 20v₁

40v₁² - 20v₁ - 40 = 0

v₁ = 1.28 m/s OR v₁ = - 0.78 m/s

The first ball is lighter so its velocity will change direction due to collision.

v₁ = - 0.78 m/s

10 = 5 (-0.78) + 7v₂

v₂ = 1.98 m/s