A sphere of mass m moving with a constant velocity collides with another stationary sphere of same mass. The ratio of velocities of two spheres after the collision will be, if coefficient of restitution is e

- Abhisar

- Stacey Warren - Expert brainly.com

Hey! We 've verified this expert answer for you, click below to unlock the details :)

- katieb

I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!

- Abhisar

@Michele_Laino

- Michele_Laino

is your collision elastic?

- Michele_Laino

or, how is defined the "coefficient of restitution"?

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- Abhisar

That's the only info given, for the sake of simplicity I tried solving it by considering the collision inelastic.

- Abhisar

Oh well, the answer is \(\frac{1-e}{1+e}\)

- Abhisar

That might help ....

- Michele_Laino

if the collision is inelastic perfectly, then the final velocity of both spheres is equal to half of the initial velocity of the colliding sphere

- Abhisar

So, you mean it can't be perfectly inelastic....Then how to find ratio of two final velocities in terms of e ?

- Michele_Laino

no, sorry I'm not sure

- Abhisar

It's ok, thnx for the effort though c:

- Michele_Laino

let's suppose that our collision is perfectly inelastic, then we have this:
|dw:1440942312488:dw|

- Michele_Laino

using the conservation of momentum we can write this:
\[\Large m{v_0} = 2mv\]
and hence the final velocity is:
\[\Large v = \frac{{{v_0}}}{2}\]

- Michele_Laino

the two spheres are sticked together, after collision

- Abhisar

Yeah i know that, but we need to find the ratio in terms of e. And I think I'll have to consider the collision elastic.

- Michele_Laino

on the other hand, if the collision is elastic, then we have this:
|dw:1440942582057:dw|

- Abhisar

But in that case one of the final velocities will become 0, isn't it?

- Abhisar

Yeah....

- Michele_Laino

namely after collision the two spheres change their velocities

- Abhisar

So basically, ratio will be 1:1 if the collision is inelastic...right?

- Abhisar

And 0 if it's elastic?

- Michele_Laino

the ratio is 1/2 if the collision is inelastic perfectly and 0 if that collision is elastic

- Abhisar

So, the value of e must be in between 0-1

- Abhisar

How, 1/2 , both the final velocities will be the same. Isn't it?

- Michele_Laino

if collision is inelastic perfectly, then we can write this:
\[\Large v = \frac{{{v_0}}}{2} \Rightarrow \frac{v}{{{v_0}}} = \frac{1}{2}\]

- Abhisar

Umm..We need to find the ratio of velocities of both the ball after collision...

- Michele_Laino

ok! then we have ratio=1

- Abhisar

Thanks, I'll try to conclude to the given answer and meanwhile if something come up in your mind then please share c:

- Michele_Laino

ok! :)

- IrishBoy123

\[mu = mv_1 + mv_2 \implies u = v_1 + v_2\]
\[e = \frac{v_2 - v_1}{u}\]
solve for \(\large \frac{v_2 }{v_1} \)

- Michele_Laino

By definition, from Wikipedia, we have that the coefficient of restitution is the ratio between the relative velocity after collision and the relative velocity before collision

- Michele_Laino

here is the article which I read:
https://en.wikipedia.org/wiki/Coefficient_of_restitution

- Michele_Laino

let's suppose that part of the kinetic energy is lost due to collision, so we can write these equations:
\[\Large \left\{ \begin{gathered}
m{v_0} = m{u_1} + m{u_2} \hfill \\
\\
\frac{{mv_0^2}}{2} = \frac{{mu_1^2}}{2} + \frac{{mu_2^2}}{2} + \varepsilon \frac{{mv_0^2}}{2} \hfill \\
\end{gathered} \right.\]
where \epsilon is such that:
\[\Large 0 \leqslant \varepsilon \leqslant 1\]
and
\[\Large \varepsilon \frac{{mv_0^2}}{2}\]
is the kinetic energy lost, due to collision, namely a fraction of the initial kinetic energy
Developing those equation I get the subsequent final velocities:
\[\Large {u_1} = {v_0}\frac{\varepsilon }{2},\quad {u_2} = {v_0}\left( {1 - \frac{\varepsilon }{2}} \right)\]
therefore, we can write this:
\[\Large \frac{{{u_2} - {u_1}}}{{{v_0}}} = 1 - \varepsilon \]

Looking for something else?

Not the answer you are looking for? Search for more explanations.