6.03f Impulse-momentum: relation

A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \text{ m s}^{-1}\) The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \text{ m s}^{-1}\) Show that the impulse on the ball has magnitude 4 N s [3 marks]

A ball of mass \(2\)kg is moving with velocity \((3\mathbf{i} - 2\mathbf{j})\)ms\(^{-1}\) when it is struck by a bat. The impulse on the ball is \((-8\mathbf{i} + 10\mathbf{j})\)Ns.

1. Find the speed of the ball immediately after the impact. [4]
2. State one modelling assumption you have used in answering part (a). [1]

A particle, of mass 2 kg, is placed at a point A on a rough horizontal surface. There is a straight vertical wall on the surface and the point on the wall nearest to A is B. The distance AB is 5 m. The particle is projected with speed 4.2 m s\(^{-1}\) along the surface from A towards B. The particle hits the wall directly and rebounds. The coefficient of friction between the particle and the surface is 0.1.

1. Determine the speed of the particle immediately before impact with the wall. [4]

The magnitude of the impulse that the wall exerts on the particle is 9.8 N s.

1. Find the speed of the particle immediately after impact with the wall. [2]

A car A of mass 1200 kg is about to tow another car B of mass 800 kg in a straight line along a horizontal road by means of a tow-rope attached between A and B. The tow-rope is modelled as being light and inextensible. Just before the tow-rope tightens, A is travelling at a speed of \(1.5 \text{ m s}^{-1}\) and B is at rest. Just after the tow-rope tightens, both cars have a speed of \(v \text{ m s}^{-1}\).

1. Find the value of \(v\). [2]
2. Calculate the magnitude of the impulse on A when the tow-rope tightens. [2]

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]

A particle of mass 5 kg is moving with velocity \(2\mathbf{i} + 5\mathbf{j}\) m s\(^{-1}\). It receives an impulse of magnitude 15 N s in the direction \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\). Find the velocity of the particle immediately afterwards. [3]

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]

Two spheres \(A\) and \(B\), of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere \(A\) has mass \(4m\) kg and sphere \(B\) has mass \(3m\) kg. Just before the collision, \(A\) has speed \(9\text{ ms}^{-1}\) and \(B\) has speed \(3.5\text{ ms}^{-1}\). Immediately after the collision, \(A\) has speed \(1.5\text{ ms}^{-1}\) in the direction of its original motion.

1. Show that the speed of \(B\) immediately after the collision is \(6.5\text{ ms}^{-1}\). [3]
2. Calculate the coefficient of restitution between \(A\) and \(B\). [3]
3. Given that the magnitude of the impulse exerted by \(B\) on \(A\) is 36 Ns, find the value of \(m\). [3]
4. Give a reason why it is not necessary to model the spheres as particles in this question. [1]

The diagram below shows two spheres \(A\) and \(B\), of equal radii, moving in the same direction on a smooth horizontal surface. Sphere \(A\), of mass \(3\) kg, is moving with speed \(4\) ms\(^{-1}\) and sphere \(B\), of mass \(2\) kg, is moving with speed \(10\) ms\(^{-1}\). \includegraphics{figure_5} Sphere \(B\) is then given an impulse after which it moves in the opposite direction with speed \(6\) ms\(^{-1}\).

1. Calculate the magnitude and direction of the impulse exerted on \(B\). [3]

Sphere \(B\) continues to move with speed \(6\) ms\(^{-1}\) so that it collides directly with sphere \(A\). The kinetic energy lost due to the collision is \(45\) J.

1. Calculate the speed of \(A\) and the speed of \(B\) immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision. [8]

A particle of mass 0.8 kg is moving in a straight line on a smooth horizontal surface with constant speed \(12 \text{ms}^{-1}\) when it is struck by a horizontal impulse. Immediately after the impulse acts, the particle is moving with speed \(9 \text{ms}^{-1}\) at an angle of 50° to its original direction of motion (see diagram). \includegraphics{figure_2} Find

1. the magnitude of the impulse, [3]
2. the angle that the impulse makes with the original direction of motion of the particle. [3]

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]

A ball of mass 0.04 kg is released from rest at a height of 1 metre above a table. It rebounds to a height of 0.81 metre.

1. Find the value of \(e\), the coefficient of restitution. [4]
2. Find the impulse on the ball when it hits the table. [3]

A light inextensible string passes over a smooth fixed pulley. Particles of mass 0.2 kg and 0.3 kg are attached to opposite ends of the string, so that the parts of the string not in contact with the pulley are vertical. The system is released from rest with the string taut.

1. Find the acceleration of the particles and the tension in the string. [6]

When the heavier particle has fallen 2.25 m it hits the ground and is brought to rest (and the string goes slack).

1. Find the speed with which it hits the ground. [2]
2. Find the magnitude of the impulse of the ground on the particle. [2]
3. If the impact between the particle and the ground lasts for 0.005 seconds, find the constant force that would be needed to bring the particle to rest. [2]

1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]

Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).

1. Find the speeds of the particles immediately after the collision. [4]
2. Determine the coefficient of restitution between the particles. [2]