Question

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

Answer 1

@Michele_Laino

Answer 2

is your collision elastic?

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

That might help ....

Answer 7

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

Answer 8

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

Answer 9

no, sorry I'm not sure

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

the two spheres are sticked together, after collision

Answer 14

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.

Answer 15

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

Answer 16

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

Answer 17

Yeah....

Answer 18

namely after collision the two spheres change their velocities

Answer 19

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

Answer 20

And 0 if it's elastic?

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

ok! then we have ratio=1

Answer 27

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

Answer 28

ok! :)

Answer 29

\[mu = mv_1 + mv_2 \implies u = v_1 + v_2\]

\[e = \frac{v_2 - v_1}{u}\]

solve for \(\large \frac{v_2 }{v_1} \)

Answer 30

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

Answer 31

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

Answer 32

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:
\[\Large \frac{{{u_2} - {u_1}}}{{{v_0}}} = 1 - \varepsilon \]