# Two small smooth particles, A and B of mass m are attached to the two ends of an inelastic string of length l and are at rest on a smooth horizontal plane . The system now moves with speed u in the...

Two small smooth particles, **A** and B of mass **m** are attached to the two ends of an inelastic string of length **l** and are at rest on a smooth horizontal plane . The system now moves with speed u in the direction AB with the string taut. The particle B, after some time, collides with a small smooth elastic particle **C** of mass **M** which is at rest on the plane. If e is the coefficient of restitution between the particles **B** and **C**,

show that the particle A collides with the particle B after a time

`(( m + M)l) / (M(1 +e)u)` from the moment of collision between B and C.

The first part is answered at http://www.enotes.com/homework-help/two-small-smooth-particles-b-mass-m-attached-two-448239

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From the previous part it has shown that after collision B has a velocity of ;

`V_B = (u(m-eM))/(M+m)`

Let us take the velocity of A relative to the particle B considering E as Earth. This means B is stationary and A is moving.

`V_(A,E) = u`

`V_(B,E) = (u(m-eM))/(M+m)`

`V(A,B) = V_(A,E)-V_(E,B)`

`V(A,B) = u-(u(m-eM))/(M+m)`

`V_(A,B) = (u(M+m-m+eM))/(M+m)`

`V_(A,B) =(Mu(1+e))/(m+M)`

Relative to B, particle A should travel a distance of l to collide with B.

Time to collide `= l/V_(A,B)`

Time to collide `= l/(Mu(1+e))/(m+M) =( (M+m)l)/(Mu(1+e))`

*So the particle A will collide with particle B, after;*

* `( (M+m)l)/(Mu(1+e))` time where B collide with C.*

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