Conservation of Momentum

• In an isolated system, the total momentum is conserved.

Example 1. In a Spinning top, total momentum = 0. For every point, there is another point on the opposite side that cancels its momentum.

 Final momentum = (M –m) (v + v’)

Thus, Mass * velocity = constant

By Newton’s Third law,

F 12 = – F21

(p₁’ – p₁)/∆t = (p₂’ – p₂) /∆t

(p₁’ – p₁) = (p₂’ – p₂)

p₁’ + p₂’ = p₁ + p₂

Conclusion: Final momentum of the system = Initial momentum of the system

Problem: A railway truck A of mass 3 * 10⁴ kg travelling at 0.6 m/s collides with another truck B of half its mass moving in the opposite direction with a velocity of 0.4 m/s. If the trucks couple automatically on collision, find the common velocity with which they move.

Solution.

m₁ = 3 * 10⁴ kg

m₂ = ½ of mass of A = 1.5 * 10⁴ kg

u₁ = 0.6 m/s

u₂ = -0.4 m/s

Before collision: m₁u₁ + m₂u₂ = 3 * 10⁴  * 0.6 + 1.5 * 10^{4 *} (-0.4)

                                                        = 1.2 * 10⁴ kg m/s

After collision: (m₁ + m₂) v = 4.5 * 10⁴ v kg m/s

As per conservation of momentum,

1.2 * 10⁴   =  4.5 * 10⁴ v

V = 1.2/ 4.5 = 0.27 m/s

Therefore, the common velocity is 0.27m/s