- #1

utkarshakash

Gold Member

- 855

- 13

## 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.

[itex]\frac{v_{2}}{vcosθ_{2}} = e[/itex]

Also,

[itex]\frac{v_{1}}{vcosθ_{1}} = e[/itex]

Applying Conservation of Momentum along X-axis

[itex]v=v_{1}cosθ_{1}+v_{2}cosθ_{2}[/itex]

Along Y- axis

[itex]v_{1}sinθ_{1}=v_{2}sinθ_{2}[/itex]

Substituting the value of v1 and v2 in the above equation

[itex]evcosθ_{1}sinθ_{1}=evcosθ_{2}sinθ_{2}[/itex]

[itex]cosθ_{1}sinθ_{1}=cosθ_{2}sinθ_{2}[/itex]

Multiplying 2 on both sides

[itex]sin2θ_{1}=sin2θ_{2}[/itex]

Rearranging and simplifying

[itex]2cos(θ_{1}+θ_{2})sin(θ_{1}-θ_{2})=0[/itex]

[itex]θ_{1}+θ_{2}=\frac{∏}{2}[/itex]

[itex]θ_{1}-θ_{2}=0[/itex]

[itex]θ_{1}=θ_{2} and θ_{1} = \frac{∏}{4}[/itex]

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

[itex]1=e(cos^{2}θ_{1}+cos^{2}θ_{2})[/itex]

[itex]e=\frac{1}{cos^{2}θ_{1}+cos^{2}θ_{2}}[/itex]

[itex]e=\frac{1}{2cos^{2}θ_{1}}[/itex]

[itex]e=1[/itex]

What's wrong here???