# How do I use the coefficient of restitution in a collision of two spheres?

#### Chemist116

The problem is as follows:

Two bodies whose masses are $m_1=\,1\,kg$ and $m_2=2\,kg$ collision with speeds $v_1=1\,\frac{m}{s}$ and $v_2=\,4\,\frac{m}{s}$ as indicated in the figure from below. If the $COR$ (coefficient of restitution) is $0.2$. Find the speed in $\frac{m}{s}$ of the second body after the collision.

The alternatives given are:

$\begin{array}{ll} 1.&0.5\,\frac{m}{s}\\ 2.&2\,\frac{m}{s}\\ 3.&1\,\frac{m}{s}\\ 4.&1.5\,\frac{m}{s}\\ 5.&3\,\frac{m}{s}\\ \end{array}$

I'm lost at this problem. Can someone help me?. The only thing I can recall is that the coefficient of restitution or COR is a relationship between the kinetic energies as follows:

$COR=\frac{\textrm{K.E after the collision}}{\textrm{K.E before the collision}}$

But I don't know how to relate this to the problem in order to solve it. Can somebody help me here?.

#### topsquark

Math Team
How about conservation of momentum? What does that tell you about the velocities?

-Dan

#### Chemist116

How about conservation of momentum? What does that tell you about the velocities?

-Dan
I'm assuming that the momentum is preserved as follows:

$p_i=p_f$

Therefore

$m_1v_1+m_2v2=m_1u_1+m_2u_2$

I'm confused if should I sum the masses of the spheres or not. (Since in inelastic collisions one sticks to the other) or could it be that this is not a perfectly inelastic collision?. I'm confused at which is which?.

$u=\textrm{final speed}$

$v=\textrm{initial speed}$

Then:

$COR=\frac{u_1-u_2}{v_1-v_2}=0.2$

Therefore:

$m_1v_1+m_2v2=m_1u_1+m_2u_2$

Here is exactly where should I put the direction, to the left? to the right?. I'll just let as it is:

$1\times 1+2\times (-4)=u_1+2u_2$

$\frac{u_1-u_2}{v_1-v_2}=\frac{u_1-u_2}{1-(-4)}=0.2$

Then all that is left to do is to solve the system of equations:

$u_1-u_2=1$

$u_1+2u_2=-7$

$u_1=\frac{-5}{3}=-1.667$

$u_2=\frac{-8}{3}=-2.667$

But this is a problem since I'm having two negative velocities and none of these checks with the alternatives. Is my analysis wrong?.

#### DarnItJimImAnEngineer

If you wrote the problem down correctly, then I agree with your answer. Both objects would travel to the left with a 1 m/s relative velocity between them. Double-check the numbers in the problem.

topsquark