1. ## collision

three spheres of mass 1kg,2kg and 3kg move in the same line with velocities 5i,i and 3i respectively. the smaller masses are the first to collide.if only one collision takes place find the maximum value for the coefficient of restitution between the smaller masses.

what ive got so far: 5(1)+1(2)=1(x)=2(y) using the law of conservation of momentum. 7=x+2y

using law if restitution -e=x-y/5-1 -4e=x=y

i also know y>x as there no more collisions. y>3 also if there are no more collisions.

2. ## Re: collision

have you worked out the initial ordering on the x axis of the spheres?

that would be my first step

if you do this you can then reduce the 3 body problem to a 2 body problem

in which the 2 smallest spheres must collide with a constraint on their post collision velocities.

see if you can get this far

3. ## Re: collision

i assume the 1kg and the 2kg will collide first as they are respectively moving with velocities 5i and i

i know y>x from my equations in OP 7-x/2=y 7-x/2>x 7/3>x i also worked out e=7-3x/8 but i dont know if its any use .i dont know where to go from here

4. ## Re: collision

This problem is perplexing me.

Would you agree that a condition for no further collisions is that, using your notation

$3 < x < y$ ?

switching notation a bit I'm going to call $x = v1_f,~y=v2_f$

conservation of momentum gets us

$v2_f = \dfrac{7-v1_f}{2}$

$e = \dfrac 1 4 (v1_f - v2_f) = \dfrac{1}{8} (3 v1_f-7)$

$v1_f = \dfrac{1}{3} (8 e+7)$

$v2_f = \dfrac{1}{3} (7-4 e)$

There is no value of $e$ such that $3 < v1_f < v2_f$

so either I'm missing something, which isn't all that unlikely, or there's something wrong with this problem.

5. ## Re: collision

shouldn't the condition be $x < y \le 3$ ?

6. ## Re: collision

$m(5) + 2m(1) = mv_1 + 2mv_2$

$7 = v_1 + 2v_2$

if there is to be only one collision, then $v_2 \le 3 \implies v_1 \ge 1$

$e \le \dfrac{\text{max rate of opening}}{\text{closing rate before collision}} = \dfrac{3-1}{5-1} = \dfrac{1}{2}$

7. ## Re: collision

Originally Posted by skeeter
shouldn't the condition be $x < y \le 3$ ?
no, they both have to be moving greater than 3 so as not to be caught by the 3kg sphere.

Edit: oh bloody hell I had the 3kg sphere on the other side.

thanks