Question

A ball with a mass of `3 kg` and velocity of `7 m/s` collides with a second ball with a mass of `5 kg` and velocity of `- 4 m/s`. If `40%` of the kinetic energy is lost, what are the final velocities of the balls?

Answer 1

After a very lengthy solution, we find

`v_(1f)=-5.2m/s`

`v_(2f)=3.32m/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 the 40% kinetic energy lost condition.

First, cons. of momentum:

`m_1v_(1i) + m_2v_(2i) = m_1v_(1f) + m_2v_(2f)`

Inserting the values we know:

`3(7) + 5(-4) = 3v_(1f) + 5v_(2f)`

`1 = 3v_(1f) + 5v_(2f)`

Now, the kinetic energy condition. The initial KE is:

`1/2m_1v_(1i)^2 + 1/2m_2v_(2i)^2`

`1/2(3)(7)^2 + 1/2(5)(-4)^2 = 73.5 + 40 = 113.5 J`

The final KE is:

`1/2m_1v_(1f)^2 + 1/2m_2v_(2f)^2`

Since 40% of the KE is lost, the final KE equals 60% of the initial

`1/2(3)v_(1f)^2 + 1/2(5)v_(2f)^2 = 113.5 xx 0.6=68.1J`

With all that complete, our two equations are:

`1 = 3v_(1f) + 5v_(2f)`

`3/2v_(1f)^2 + 5/2v_(2f)^2 = 68.1`

Rewrite the first equation as `v_(1f)=(1-5v_(2f))/3`

Substitute this value into the second equation:

`3/2((1-5v_(2f))/3)^2 + 5/2v_(2f)^2 = 68.1`

Now, we must solve this equation. It will be simpler if we multiply every term by 18, to eliminate the denominators:

`3(1-5v_(2f))^2 + 45v_(2f)^2 = 1225.8`

`3(1-10v_(2f)+25v_(2f)^2) + 45v_(2f)^2 = 1225.8`

`3-30v_(2f)+75v_(2f)^2 + 45v_(2f)^2 = 1225.8`

`120v_(2f)^2-30v_(2f)-1222.8=0`

Use the quadratic formula to solve for v_(2f)

`v_(2f)= (30+- sqrt((-30)^2 - 4(120)(-1222.8)))/(2(120))`

`= (30+-766.7)/240`

There are two answers:

`v_(2f)= (30+766.7)/240=3.32m/s` and

`v_(2f)= (30-766.7)/240=-3.07m/s`

Substituting this answer back into `v_(1f)=(1-5v_(2f))/3`

we get `v_(1f)=(1-5(3.32))/3=-5.2m/s`

and `v_(1f)=(1-5(-3.07))/3=5.45m/s`

We must reject the second 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 , but somehow the balls lost 40% of their KE.

So, the only acceptable answers are

`v_(1f)=-5.2m/s`

`v_(2f)=3.32m/s`