A ball with a mass of 5kg is rolling at 18ms and elastically collides with a resting ball with a mass of 2kg. What are the post-collision velocities of the balls?

1 Answer
Jan 6, 2017

After collision, the 5-kg ball moves in the same direction at 7.71m/s while the 2-kg ball also moves in this direction at 25.7 m/s.

Explanation:

This is a lengthy problem that asks us to solve for two unknowns (the two final velocities). To do this, we must generate two equations involving these two unknowns, and solve them simultaneously.

One equation will come from conservation of momentum, the other will come from kinetic energy conservation.

First, cons. of momentum:

m1v1i+m2v2i=m1v1f+m2v2f

Inserting the values we know:

5(18)+2(0)=5v1f+2v2f

90=5v1f+2v2f

Now, kinetic energy conservation.

12m1v21i+12m2v22i=12m1v21f+12m2v22f

12(5)(18)2+12(2)(0)2=12(5)v21f+12(2)v22f

810=2.5v21f+(1)v22f

With all that complete, our two equations are:

90=5v1f+2v2f

810=2.5v21f+(1)v22f

Rewrite the first equation as v2f=905v1f2

Substitute this value into the second equation:

2.5v21f+(905v1f2)2=810

Now, we must solve this equation. It will be simpler if we multiply every term by 4, to eliminate the denominators:

10v21f+(905v1f)2=3240

10v21f+(8100900v1f+25v21f)=3240

35v21f900v1f+4860=0

Use the quadratic formula to solve for v_(1f)

v1f=900±(900)24(35)(4860)2(35)

=900±36070

There are two answers:

v1f=900+36070=18ms and

v1f=90036070=7.71ms

Substituting each answer back into v2f=905v1f2

we get v2f=905(18)2=0ms

and v2f=905(7.71)2=25.7ms

We must reject the first answer in each case, as this would result in both balls continuing in their original directions. This could only happen if there was no collision.

So, the only acceptable answers are

v1f=7.71ms

v2f=25.7ms