# Find the coefficient of restitution

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## Homework Statement

Two equal balls are in contact on a table. A third equal ball strikes them simultaneously and remains at rest after the impact. Show that the coefficient of restitution is 2/3.
I have attached an image for clarity of problem. Please open it.

## Homework Equations

Conservation of Momentum

## The Attempt at a Solution

Let the mass of balls be m and coefficient of restitution be e.

$\frac{v_{2}}{vcosθ_{2}} = e$

Also,
$\frac{v_{1}}{vcosθ_{1}} = e$

Applying Conservation of Momentum along X-axis

$v=v_{1}cosθ_{1}+v_{2}cosθ_{2}$
Along Y- axis
$v_{1}sinθ_{1}=v_{2}sinθ_{2}$

Substituting the value of v1 and v2 in the above equation
$evcosθ_{1}sinθ_{1}=evcosθ_{2}sinθ_{2}$
$cosθ_{1}sinθ_{1}=cosθ_{2}sinθ_{2}$

Multiplying 2 on both sides
$sin2θ_{1}=sin2θ_{2}$
Rearranging and simplifying
$2cos(θ_{1}+θ_{2})sin(θ_{1}-θ_{2})=0$
$θ_{1}+θ_{2}=\frac{∏}{2}$
$θ_{1}-θ_{2}=0$

$θ_{1}=θ_{2} and θ_{1} = \frac{∏}{4}$

Now substituting the value of θ1 in equation of momentum along X-axis

$1=e(cos^{2}θ_{1}+cos^{2}θ_{2})$
$e=\frac{1}{cos^{2}θ_{1}+cos^{2}θ_{2}}$
$e=\frac{1}{2cos^{2}θ_{1}}$
$e=1$

What's wrong here??? #### Attachments

I will say first to reconsider your expressions for the coefficients of restitution. Also check the momentum conservation. Furthermore it may help to consider energy conservation as well. As this gives a further restraint on the values in the problem.

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I will say first to reconsider your expressions for the coefficients of restitution. Also check the momentum conservation. Furthermore it may help to consider energy conservation as well. As this gives a further restraint on the values in the problem.
energy conservation principle can't be applied here because the collision is not elastic. I also suspect something wrong in my expressions for coefficient of restitution. But I can't figure out what is it. Also I don't think that momentum conservation equations are wrong.

Well if it is indeed an inelastic collision than you are correct. It was not specified in your problem statement. You're off by a minus sign in one of your momentum conservation equations. The definition of the coefficient of restitution is fractional value of the ratio of speeds before and after a collision. You should be able to find the error by just considering this definition and looking at your expressions for, e.

If the 3 balls are identical and the collisions occur simultaneously. You can immediately deduce that.
The lines of impacts of both collisions pass through the centres and therefore,
$$\theta_1=\theta_2=\frac{\pi}{6}\\ and\\ v_1=v_2$$
Then your answer is therefore $e=\frac{2}{3}$

#### Attachments

Last edited:
Gold Member
If the 3 balls are identical and the collisions occur simultaneously. You can immediately deduce that.
The lines of impacts of both collisions pass through the centres and therefore,
$$\theta_1=\theta_2=\frac{\pi}{6}\\ and\\ v_1=v_2$$
Then your answer is therefore $e=\frac{2}{3}$
Hey, why didn't I think it earlier!!!!! Thank You.