Bouncing Particles on a smooth table

[Bucky]

"Particles A, B and C , each of mass m, lie at rest in a straight line in the order stated. A is projected directly towards B with velocity u. The coefficient of restitution is 0.5 in each impact that follows. Show that there will be three impacts in total and find the final velocities of each particle."

I have 'an' answer for the first part:-

A and B Collide

Velocity of separation $$= e (u_1 - u_2)<br /> =0.5(u-0)<br /> =0.5u$$
$$v_1 + v_2 = u // <br /> v_1 = u - v_2$$

$$v_2 - v_1 = e(u_1 - u_2)//<br /> v_2 - u + v_2 = 0.5u//<br /> 2v_2 = 1.5u//<br /> v_2 = 0.75u$$
$$v_1 = u - v_2<br /> v_1 = u - 0.75u<br /> v_1 = 0.25u$$

is this the right way to do this question? the problem comes later when the moving particles hit each other.

 

[wais]

Yes, your approach to finding the velocities after the first collision between A and B is correct. However, there are a few things to consider for the subsequent collisions.

First, after the first collision, particle B will be moving with a velocity of 0.75u and particle A will be moving with a velocity of 0.25u. This means that when A reaches B for the second collision, their velocities will be different from the initial conditions given in the problem.

Second, since the particles are on a smooth table, there will be no external forces acting on them to change their velocities between collisions. This means that the velocities of particles A and B will remain constant throughout the collisions.

Therefore, for the second collision, the velocities of particles A and B will be 0.25u and 0.75u respectively. Using the same approach as before, we can find the velocities after the second collision to be 0.125u and 0.625u for particles A and B respectively.

For the third collision, the velocities of particles A and B will be 0.125u and 0.625u respectively. Using the same approach again, we can find the final velocities to be 0.0625u and 0.5625u for particles A and B respectively.

Since particle C is not involved in any of the collisions, its final velocity will remain 0.

Therefore, there will be three impacts in total and the final velocities of particles A, B, and C will be 0.0625u, 0.5625u, and 0 respectively.