Range of coefficient of restitution

1 Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2 m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4 u\) and \(3 u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision.

1. A particle \(P\) of mass \(2 m\) and a particle \(Q\) of mass \(5 m\) are moving along the same straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The direction of motion of \(Q\) is reversed by the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\). Given that \(e = \frac { 1 } { 3 }\)
2. show that the kinetic energy lost in the collision is \(\frac { 40 m u ^ { 2 } } { 7 }\).
3. Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if \(e > \frac { 1 } { 3 }\)

Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4u\) and \(3u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision. [5]

Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(5m\), \(5m\) and \(3m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).

1. Show that the speed of \(A\) after its collision with \(B\) is \(\frac{1}{2}u(1 - e)\) and find the speed of \(B\). [3]

Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.

1. Find the set of possible values of \(e\). [6]

Two particles \(A\) and \(B\) have masses \(3m\) and \(2m\) respectively. Initially \(A\) is at rest and \(B\) is moving with speed \(u\) in a straight line towards \(A\). The coefficient of restitution between the particles is \(e\).

1. Find the speeds of the particles immediately after the collision.
2. Find the condition on \(e\) for \(A\) to be moving faster than \(B\) after the collision.

[8]

Three particles \(A\), \(B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5u\) and \(4u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3u\). In the subsequent motion, \(A\), \(B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. [7]

Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),

1. find the set of possible values of \(e\). [4]

A particle \(A\) of mass \(2m\) is moving with speed \(2u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Show that the speed of \(B\) after the collision is \(\frac{3}{2}u\). [6]

1. Find the speed of \(A\) after the collision. [2]

Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,

1. find the range of possible values for \(e\). [8]

A particle \(P\) of mass \(2m\) is moving with speed \(2u\) in a straight line on a smooth horizontal plane. A particle \(Q\) of mass \(3m\) is moving with speed \(u\) in the same direction as \(P\). The particles collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac{3}{2}u\). [5]
2. Find the total kinetic energy lost in the collision. [5]

After the collision between \(P\) and \(Q\), the particle \(Q\) collides directly with a particle \(R\) of mass \(m\) which is at rest on the plane. The coefficient of restitution between \(Q\) and \(R\) is \(e\).

1. Calculate the range of values of \(e\) for which there will be a second collision between \(P\) and \(Q\). [7]

A particle \(A\) of mass \(2m\) is moving with speed \(2u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Show that the speed of \(B\) after the collision is \(\frac{5}{3}u\). [6]
2. Find the speed of \(A\) after the collision. [2]

Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,

1. find the range of possible values for \(e\). [8]

Two small uniform smooth spheres A and B, of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere A has speed 2 m s\(^{-1}\) and B has speed 1 m s\(^{-1}\) before they collide. The coefficient of restitution between A and B is \(e\).

1. Show that the velocity of B after the collision, in the original direction of motion of A, is \(\frac{1}{5}(1 + 6e)\) m s\(^{-1}\) and find a similar expression for the velocity of A after the collision. [5]
2. The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B.
   1. In the collision between A and B the spheres coalesce to form a combined body C. State the speed of C after the collision. [1]
   2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of \(e\). [2]
   3. The total loss in kinetic energy due to the collision is 3 J. Determine the value of \(e\). [4]