A ball with a mass of 1 kg moving at a speed of 4 m / s collides with a ball with a mass of 8 kg

A ball with a mass of 1 kg moving at a speed of 4 m / s collides with a ball with a mass of 8 kg, which is moving towards it at a speed of 1 m / s. Considering the impact to be central and elastic, determine the speed of movement of the balls after the impact.

m1 = 1 kg.

V1 = 4 m / s.

m2 = 8 kg.

V2 = 1 m / s.

V1 “-?

V2 “-?

Let us write down the law of conservation of total mechanical energy and the law of conservation of momentum.

m 1 * V1 ^ 2/2 + m2 * V2 ^ 2/2 = m 1 * V1 “2/2 + m2 * V2” 2/2.

m 1 * V1 ^ 2 – m 1 * V1 “2 = m2 * V2” 2 – m2 * V2 ^ 2.

m 1 * (V1 ^ 2 – V1 “2) = m2 * (V2” 2 – V2 ^ 2).

m 1 * (V1 – V1 “) * (V1 + V1”) = m2 * (V2 “- V2) * (V2” + V2).

The impulse conservation law: m 1 * V1 – m2 * V2 = m2 * V2 “- m 1 * V1”.

m 1 * V1 + m 1 * V1 “= m2 * V2 + m2 * V2”.

m 1 * (V1 + V1 “) = m2 * (V2 + V2”).

Divide the first equation by the second:

m 1 * (V1 – V1 “) * (V1 + V1”) / m 1 * (V1 + V1 “) = m2 * (V2” – V2) * (V2 “+ V2) / m2 * (V2 + V2” ).

V1 – V1 “= V2” – V2.

V2 “= V1 – V1” + V2.

m 1 * V1 + m 1 * V1 “= m2 * V2 + m2 * (V1 – V1” + V2).

m 1 * V1 + m 1 * V1 “= m2 * V2 + m2 * V1 – m2 * V1” + m2 * V2.

m1 * V1 “+ m2 * V1” = 2 * m2 * V2 + m2 * V1 – m 1 * V1.

V1 “= (2 * m2 * V2 + m2 * V1 – m 1 * V1) / (m1 + m2).

V1 “= (2 * 8 kg * 1 m / s + 8 kg * 4 m / s – 1 kg * 4 m / s) / (1 kg + 8 kg) = 4.8 m / s.

V2 “= 4 m / s – 4.8 m / s + 1 m / s = 0.2 m / s.

Answer: V1 “= 4.8 m / s, V2” = 0.2 m / s.