6.03i Coefficient of restitution: e

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]

Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u_1 \mathbf{i} + u_2 \mathbf{j}\) towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf{i}\). [1]
   2. Determine this impulse in terms of \(\lambda\), \(m\), \(e\) and \(u_1\). [6]

The loss in kinetic energy due to the collision between A and B is \(\frac{1}{8}mu_1^2\).

1. Determine the range of possible values of \(\lambda\). [6]

A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of \(14 \text{ m s}^{-1}\). At the same instant a particle Q of mass 8 kg is released from rest 5 m vertically above P. During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is \(\frac{11}{14}\). Determine the time between this collision and P subsequently hitting the ground. [10]

Two small uniform discs A and B, of equal radius, have masses 3 kg and 5 kg respectively. The discs are sliding on a smooth horizontal surface and collide obliquely. The contact between the discs is smooth and A is stationary after the collision. Immediately before the collision B is moving with speed \(2 \text{ m s}^{-1}\) in a direction making an angle of \(60°\) with the line of centres, XY (see diagram below). \includegraphics{figure_12}

1. Explain how you can tell that A must have been moving along XY before the collision. [1]

The coefficient of restitution between A and B is 0.8.

1. • Determine the speed of A immediately before the collision. • Determine the speed of B immediately after the collision. [7]
2. Determine the angle turned through by the direction of B in the collision. [3]

Disc B subsequently collides with a smooth wall, which is parallel to XY. The kinetic energy of B after the collision with the wall is 95% of the kinetic energy of B before the collision with the wall.

1. Determine the coefficient of restitution between B and the wall. [4]

A small ball of mass \(m\) kg is held at a height of \(78.4\) m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is \(e\). The ball continues to bounce until it comes to rest after \(6\) seconds.

1. Determine the value of \(e\). [8]
2. Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is \(23.52\) Ns, determine the value of \(m\). [2]

In this question take \(g = 10\). A smooth ball of mass 0.1 kg is projected from a point on smooth horizontal ground with speed 65 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \frac{3}{4}\). While it is in the air the ball is modelled as a particle moving freely under gravity. The ball bounces on the ground repeatedly. The coefficient of restitution for the first bounce is 0.4.

1. Show that the ball leaves the ground after the first bounce with a horizontal speed of 52 m s\(^{-1}\) and a vertical speed of 15.6 m s\(^{-1}\). Explain your reasoning carefully. [4]
2. Calculate the magnitude of the impulse exerted on the ball by the ground at the first bounce. [2]

Each subsequent bounce is modelled by assuming that the coefficient of restitution is 0.4 and that the bounce takes no time. The ball is in the air for \(T_1\) seconds between projection and bouncing the first time, \(T_2\) seconds between the first and second bounces, and \(T_n\) seconds between the \((n-1)\)th and \(n\)th bounces.

1. 1. Show that \(T_1 = \frac{39}{5}\). [2]
   2. Find an expression for \(T_n\) in terms of \(n\). [2]
2. According to the model, how far does the ball travel horizontally while it is still bouncing? [3]
3. According to the model, what is the motion of the ball after it has stopped bouncing? [1]

Two objects, \(A\) of mass 18 kg and \(B\) of mass 7 kg, are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of \(4\text{ ms}^{-1}\) and in the same direction. Object \(B\) collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of \(3\text{ ms}^{-1}\). Subsequently, the two objects \(A\) and \(B\) collide directly. The coefficient of restitution between the two objects is \(\frac{5}{7}\).

1. Find the coefficient of restitution between \(B\) and the wall. [1]
2. Determine the speed of \(A\) and the speed of \(B\) immediately after the two objects collide. [7]
3. Calculate the impulse exerted by \(A\) on \(B\) due to the collision and clearly state its units. [2]
4. Find the loss in energy due to the collision between \(A\) and \(B\). [2]
5. State the direction of motion of \(A\) relative to the wall after the collision with \(B\). [1]

\includegraphics{figure_2} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W_1\) and \(W_2\) are two fixed parallel vertical walls. The walls are \(3\) metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leq 3\), from \(W_1\). At time \(t = 0\), the particle is projected from \(O\) towards \(W_1\) with speed \(u\text{ms}^{-1}\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac{2}{3}\). The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.

1. Show that \(T = \frac{45 - 5d}{4u}\). [6]
2. The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer. [2]

Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]

After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).

1. Determine the range of possible values of \(e\). [4]

A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(BC = l\), by a light inextensible string of length \(l\). \(A\) is released from rest with the string \(OA\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics{figure_8} \(A\) moves in a vertical plane perpendicular to \(CB\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.

1. Show that, on the next occasion that \(A\) comes to rest, the string \(OA\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac{3 + \cos \theta}{4}\). [9]

\(A\) and \(B\) collide again when \(AO\) is next vertical.

1. Find the percentage of the original energy of the system that remains immediately after this collision. [5]
2. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision. [1]
3. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. [1]

\includegraphics{figure_8} A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]