1. ## collision

2 smooth spheres of masses m and 3m move with speeds u and v in opposite directions . the smaller mass is brought to rest .i show u greater than or equal to 3v

im trying to working this out using the 2 equations m(u)+3m(-v)=m(0)+3m(x) where x is the final velocity of the 3m mass . im also using 0-q/u+v =-e

2. ## Re: collision

are you saying that the small mass ends up with velocity zero after the collision due to conservation of momentum?

3. ## Re: collision

There is the question. I should of put up the first part of the question though I know how to work it out. I'm stuck on part ii. the second ball has zero velocity after the collision but the other mass is moving

4. ## Re: collision

initially sphere 1 has momentum

$p1_i= m u$ and likewise sphere 2

$p2_i = -3mv$

The initial system momentum is

$p_i =p1_i + p2_i = m(u-3v)$, and the initial relative velocity is

$\Delta v_i = u+v$

They collide and sphere 1 is brought to a stop. Sphere 2 moves off with some as yet unknown velocity, $v_f$ and momentum is conserved

$m(u-3v) = 0 + 3mv_f$

$v_f = \dfrac{u-3v}{3}$, and as sphere 1 has zero velocity the final relative velocity is just $\Delta v_f = v_f$

$e = \dfrac{\Delta v_f}{\Delta v_i} = \dfrac{u-3v}{3(u+v)} = \dfrac{u-3v}{3u+3v}$

as for the second part it looks like they simply insist $e \geq 0$ and thus $u\geq 3v$

here's what Wiki has to say about negative coefficients of restitution

"e < 0: A COR less than zero would represent a collision in which the separation velocity of the objects has the same direction (sign) as the closing velocity, implying the objects passed through one another without fully engaging. This may also be thought of as an incomplete transfer of momentum. An example of this might be a small, dense object passing through a large, less dense one – e.g., a bullet passing through a target, or a motorcycle passing through a motor home or a wave tearing through a dam."

Nothing like this is going on. The two solid spheres collide without deformation so it must be that $e \geq 0$

5. ## Re: collision

thanks and is e not e>0 and not e≥0. it cant be equal to zero or both masses would of come to rest

6. ## Re: collision

Originally Posted by markosheehan
thanks and is e not e>0 and not e≥0. it cant be equal to zero or both masses would of come to rest
you're right since the first sphere comes to rest.