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Abhisar
 one year ago
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
 one year ago
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

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Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1is your collision elastic?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1or, how is defined the "coefficient of restitution"?

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0That's the only info given, for the sake of simplicity I tried solving it by considering the collision inelastic.

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0Oh well, the answer is \(\frac{1e}{1+e}\)

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1if 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
 one year ago
Best ResponseYou've already chosen the best response.0So, you mean it can't be perfectly inelastic....Then how to find ratio of two final velocities in terms of e ?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1no, sorry I'm not sure

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0It's ok, thnx for the effort though c:

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1let's suppose that our collision is perfectly inelastic, then we have this: dw:1440942312488:dw

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1using 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
 one year ago
Best ResponseYou've already chosen the best response.1the two spheres are sticked together, after collision

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0Yeah 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
 one year ago
Best ResponseYou've already chosen the best response.1on the other hand, if the collision is elastic, then we have this: dw:1440942582057:dw

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0But in that case one of the final velocities will become 0, isn't it?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1namely after collision the two spheres change their velocities

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0So basically, ratio will be 1:1 if the collision is inelastic...right?

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0And 0 if it's elastic?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1the ratio is 1/2 if the collision is inelastic perfectly and 0 if that collision is elastic

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0So, the value of e must be in between 01

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0How, 1/2 , both the final velocities will be the same. Isn't it?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1if collision is inelastic perfectly, then we can write this: \[\Large v = \frac{{{v_0}}}{2} \Rightarrow \frac{v}{{{v_0}}} = \frac{1}{2}\]

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0Umm..We need to find the ratio of velocities of both the ball after collision...

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1ok! then we have ratio=1

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.0Thanks, I'll try to conclude to the given answer and meanwhile if something come up in your mind then please share c:

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.0\[mu = mv_1 + mv_2 \implies u = v_1 + v_2\] \[e = \frac{v_2  v_1}{u}\] solve for \(\large \frac{v_2 }{v_1} \)

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1By 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
 one year ago
Best ResponseYou've already chosen the best response.1here is the article which I read: https://en.wikipedia.org/wiki/Coefficient_of_restitution

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1let'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: \[\Large \frac{{{u_2}  {u_1}}}{{{v_0}}} = 1  \varepsilon \]
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