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

Jun 2017
276
6
Lima, Peru
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
May 2013
2,381
993
The Astral plane
How about conservation of momentum? What does that tell you about the velocities?

-Dan
 
Jun 2017
276
6
Lima, Peru
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?.
 
Jun 2019
493
260
USA
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.
 
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