Direct collision, find velocities

1 Two uniform small smooth spheres, \(A\) and \(B\), of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere \(A\) receives an impulse of magnitude 8 N s in the direction \(A B\). The coefficient of restitution between the spheres is \(e\). Find, in terms of \(e\), the speeds of \(A\) and \(B\) after \(A\) collides with \(B\). Given that the spheres move in opposite directions after the collision, show that \(e > \frac { 2 } { 3 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_305_1301_1708_424} Two small spheres \(A\) and \(B\) of masses 2 kg and 3 kg respectively lie at rest on a smooth horizontal platform which is fixed at a height of 4 m above horizontal ground (see diagram). Sphere \(A\) is given an impulse of 6 N s towards \(B\), and \(A\) then strikes \(B\) directly. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that the speed of \(B\) after it has been hit by \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\). Sphere \(B\) leaves the platform and follows the path of a projectile.
2. Calculate the speed and direction of motion of \(B\) at the instant when it hits the ground.

4 A smooth sphere \(A\), of mass \(m\), is moving with speed \(4 u\) in a straight line on a smooth horizontal table. A smooth sphere \(B\), of mass \(3 m\), has the same radius as \(A\) and is moving on the table with speed \(2 u\) in the same direction as \(A\). \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-10_289_780_493_625} The sphere \(A\) collides directly with sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speeds of \(A\) and \(B\) immediately after the collision.
2. Show that the speed of \(B\) after the collision cannot be greater than \(3 u\).
3. Given that \(e = \frac { 2 } { 3 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) in the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

5 Point A lies 20 m vertically below a point B . A particle P of mass 4 m kg is projected upwards from A , at a speed of \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same time, a particle Q of mass \(m \mathrm {~kg}\) is released from rest at point B . The particles collide directly, and it is given that the coefficient of restitution in the collision between P and Q is 0.6 .

1. Show that, immediately after the collision, P continues to travel upwards at \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and determine, at this time, the corresponding velocity of Q . In another situation, a particle of mass \(3 m \mathrm {~kg}\) is released from rest and falls vertically. After it has fallen 10 m , it explodes into two fragments. Immediately after the explosion, the lower fragment, of mass \(2 m \mathrm {~kg}\), moves vertically downwards with speed \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the upper fragment, of mass \(m \mathrm {~kg}\), moves vertically upwards with speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Given that, in the explosion, the kinetic energy of the system increases by \(72 \%\), show that \(2 v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } = 1011.36\).
3. By finding another equation connecting \(v _ { 1 }\) and \(v _ { 2 }\), determine the speeds of the fragments immediately after the explosion.

4 Two smooth spheres \(A\) and \(B\) of equal radius are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(m\) and \(4 m\) respectively.
The coefficient of restitution between the spheres is \(e\).
The spheres are projected directly towards each other, each with speed \(u\), and subsequently collide. 4

1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
2. Find the speed of \(A\) in terms of \(u\) and \(e\).
   4
3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
   4
4. The magnitude of the impulse on \(B\) due to the collision is \(I\).
   Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$