A-level Physics (Advancing Physics)/Conservation of Momentum

Momentum is the product of the mass of an object and its velocity. It is usually denoted p:

$p=mv$,

where m is mass, and v is velocity. The total momentum in a closed system is always conserved. This fact is useful, since it allows us to calculate velocities and masses in collisions.

Collisions

Let us consider a basic example: a ball of mass M collides with velocity u with a stationary ball of mass m. The stationary ball has no momentum before the collision, and the moving ball has momentum Mu. This must equal the momentum of both balls after the collision. If we let their velocities be v₁ and v₂:

$Mu=Mv_{1}+Mv_{2}$

At this point, we would need to know one of the velocities afterwards in order to calculate the other.

Alternatively, we could have one ball of mass M colliding with another ball of mass m, with both balls moving in opposite directions with velocities u₁ and u₂ respectively. If we define the direction of motion of the ball with mass M as the positive direction:

$Mu_{1}-mu_{2}=Mv_{1}+Mv_{2}$

We do not need to worry about the signs on the right-hand side: they will take care of themselves. If one of our velocities turns out to be negative, we know that it is in the opposite direction to u₁.

Elasticity

The collisions in a Newton's cradle are almost perfectly elastic.

Although momentum within a closed system is always conserved, kinetic energy does not have to be. If kinetic energy is conserved in a collision, then it is known as a perfectly (or totally) elastic collision. If it is not conserved, then the collision is inelastic. If the colliding particles stick together, then a totally inelastic collision has occurred. This does not necessarily mean that the particles have stopped. In a totally inelastic collision, the two particles become one, giving the equation:

$Mu_{1}+mu_{2}=(M+m)v$

$v={\frac {Mu_{1}+mu_{2}}{M+m}}$

Explosions

In an explosion, two particles which are stuck together are no longer stuck together, and so gain separate velocities:

$(M+m)u=Mv_{1}+mv_{2}$

Questions

1. A ball of mass 0.5 kg collides with a stationary ball of 0.6 kg at a velocity of 3ms⁻¹. If the stationary ball moves off at a speed of 2ms⁻¹, what is the new velocity of the first ball?

2. Two balls are moving in opposite directions with velocities 5ms⁻¹ and 10ms⁻¹. They collide, and move off in opposite directions with new velocities of 7.5ms⁻¹ each. If the mass of the first ball was 1.25 kg, what is the mass of the second ball?

3. A totally elastic collision occurs between two balls of equal mass. One of the balls is stationary. What happens?

4. A particle explodes to become two particles with masses 1 kg and 2 kg. The 1 kg particle moves with velocity 45ms⁻¹. With what velocity does the other particle move?

5. A 3 kg ball moving at 3ms⁻¹ collides with a 5 kg ball moving at -5ms⁻¹. The collision is perfectly elastic. What are the new velocities of the balls?

6. A ball collides with a wall, and rebounds at the same velocity. Why doesn't the wall move?

Worked Solutions