A ball with a mass of 3kg is rolling at 4ms1 and elastically collides with a resting ball with a mass of 4kg. What are the post-collision velocities of the balls?

1 Answer
Jan 15, 2016

In an elastic collision, both momentum and kinetic energy are conserved. The velocity of the 3kg ball is 0.9or2.7ms1 and the velocity of the 4kg ball is 2.7or1.1ms1.

Explanation:

Momentum is conserved in all collisions. Kinetic energy is conserved in elastic collisions but not in inelastic or partially elastic collisions.

The initial momentum of the whole system is p=mv for the 3kg ball: the 4kg ball has zero momentum because its has zero velocity.

p=mv=34=12kgms1

The total momentum after the collision will be the same after the collision. If we call the 3kg ball 1 and the 4kg ball 2, the final momentum will be given by:

p=m1v1+m2v2=3v1+4v2=12 (call this Equation 1)

The total kinetic energy before the collision will be Ek=12mv2 for the 3kg ball only - the 4kg ball has zero kinetic energy because it has zero velocity.

Ek=12mv2=12342=24J

Since kinetic energy is conserved, the final kinetic energy will be the same, and will be given by:

Ek=12m1v21+12m2v22=32v21+42v22=24

Multiply both sides by 2 to make it tidier:

3v21+4v22=48 (call this Equation 2)

We now have two equations and two unknowns, so we can solve them as simultaneous equations. Rearranging Equation 1 to express v2 in terms of v1:

v2=123v14

Substituting this into Equation 2:

3v21+4(123v14)2=48

I'll leave the algebra as an exercise for the reader, but this solves to give v1=0.9ms1or2.5ms1. Substituting this back into Equation 1 gives v2=2.7ms1or1.1ms1, respectively.

Substituting these values should confirm that both momentum and kinetic energy were conserved.