Particle brought to rest by collision

6. Three particles \(P , Q\) and \(R\) have masses \(3 m , k m\) and 7.5m respectively. The three particles lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 9 }\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 10 u } { 3 ( 3 + k ) }\).
2. Find the range of values of \(k\) for which the direction of motion of \(P\) is reversed as a result of the collision. Following the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\). Given that \(k = 7\) and that \(Q\) is brought to rest by the collision with \(R\),
3. find the total kinetic energy lost in the collision between \(Q\) and \(R\).

6. A smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another smooth sphere \(B\) of mass \(3 m\), which is at rest on the table. The coefficient of restitution between \(A\) and \(B\) is \(e\). The spheres have the same radius and are modelled as particles.

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } ( 1 + e ) u\).
2. Find the speed of \(A\) immediately after the collision. Immediately after the collision the total kinetic energy of the spheres is \(\frac { 1 } { 6 } m u ^ { 2 }\).
3. Find the value of \(e\).
4. Hence show that \(A\) is at rest after the collision.

5. A smooth sphere \(S\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal plane. The sphere \(S\) collides with another smooth sphere \(T\), of equal radius to \(S\) but of mass \(k m\), moving in the same straight line and in the same direction with speed \(\lambda u , 0 < \lambda < \frac { 1 } { 2 }\). The coefficient of restitution between \(S\) and \(T\) is \(e\). Given that \(S\) is brought to rest by the impact,

1. show that \(e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }\).
2. Deduce that \(k > 1\).

4 Two smooth spheres \(P\) and \(Q\), of equal radius, have masses \(m\) and \(3 m\) respectively. They are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(P\) has speed \(u\) and collides directly with sphere \(Q\) which has speed \(k u\), where \(0 < k < 1\). Sphere \(P\) is brought to rest by the collision. Show that the coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 k + 1 } { 3 ( 1 - k ) }\). One third of the total kinetic energy of the spheres is lost in the collision. Show that $$k = \frac { 1 } { 3 } ( 2 \sqrt { } 3 - 3 )$$

9 A particle of mass 0.5 kg moving on a smooth horizontal plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides directly with another particle of mass \(k \mathrm {~kg}\) (where \(k\) is a constant) which is at rest. After the collision the first particle comes to rest but the second particle moves off with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(v\) in terms of \(k\) and \(u\).
2. The coefficient of restitution between the two particles is \(e\). Find \(e\) in terms of \(k\) only.
3. Show that \(k \geqslant \frac { 1 } { 2 }\).

Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac{1}{4}u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). [6] Deduce that \(\alpha \geqslant 2\). [2]

Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.

1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]

Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).

1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]