6.03f Impulse-momentum: relation

5 Two particles, \(P\) and \(Q\), of masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively, are held at rest in the same vertical line. The heights of \(P\) and \(Q\) above horizontal ground are 1 m and 2 m respectively. \(P\) is projected vertically upwards with speed \(2 \mathrm {~ms} ^ { - 1 }\). At the same instant, \(Q\) is released from rest.

1. Find the speed of each particle immediately before they collide.
2. It is given that immediately after the collision the downward speed of \(Q\) is \(3.5 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(P\) at the instant that it reaches the ground.

3 Three small smooth spheres \(A , B\) and \(C\) of equal radii and of masses \(4 \mathrm {~kg} , 2 \mathrm {~kg}\) and 3 kg respectively, lie in that order in a straight line on a smooth horizontal plane. Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collison with \(B\), sphere \(A\) continues to move in the same direction but with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(B\) after this collison.
   Sphere \(B\) collides with \(C\). In this collison these two spheres coalesce to form an object \(D\).
2. Find the speed of \(D\) after this collision.
3. Show that the total loss of kinetic energy in the system due to the two collisions is 38.4 J .

1. Two particles \(B\) and \(C\) have mass \(m \mathrm {~kg}\) and 3 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. The two particles collide directly. Immediately before the collision, the speed of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(C\) is reversed and the direction of motion of \(B\) is unchanged. Immediately after the collision, the speed of \(B\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find

1. the value of \(m\),
2. the magnitude of the impulse received by \(C\).

1. A railway truck \(P\), of mass \(m \mathrm {~kg}\), is moving along a straight horizontal track with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Truck \(P\) collides with a truck \(Q\) of mass 3000 kg , which is at rest on the same track. Immediately after the collision the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is reversed by the collision.

Modelling the trucks as particles, find

1. the magnitude of the impulse exerted by \(P\) on \(Q\),
2. the value of \(m\).

1. Two small balls \(A\) and \(B\) have masses 0.5 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) immediately after the collision is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of the motion of \(A\) is unchanged as a result of the collision.

By modelling the balls as particles, find

1. the speed of \(B\) immediately after the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision.

2. Two particles \(A\) and \(B\) have mass 0.12 kg and 0.08 kg respectively. They are initially at rest on a smooth horizontal table. Particle \(A\) is then given an impulse in the direction \(A B\) so that it moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards \(B\).

1. Find the magnitude of this impulse, stating clearly the units in which your answer is given.
   (2) Immediately after the particles collide, the speed of \(A\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its direction of motion being unchanged.
2. Find the speed of \(B\) immediately after the collision.
3. Find the magnitude of the impulse exerted on \(A\) in the collision.

8. A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road. The truck has mass 1200 kg and the car has mass 800 kg . The truck is connected to the car by a horizontal rope which is modelled as light and inextensible. The truck's engine provides a constant driving force of 2400 N . The resistances to motion of the truck and the car are modelled as constant and of magnitude 600 N and 400 N respectively. Find

1. the acceleration of the truck and the ear,
2. the tension in the rope. When the truck and car are moving at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The engine of the truck provides the same driving force as before. The magnitude of the resistance to the motion of the truck remains 600 N .
3. Show that the truck reaches a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) approximately 6 s earlier than it would have done if the rope had not broken. \section*{END}

2. Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. They are moving in opposite directions on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As a result of the collision, the direction of motion of \(B\) is reversed and its speed immediately after the collision is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) immediately after the collision, stating clearly whether the direction of motion of \(A\) is changed by the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision, stating clearly the units in which your answer is given.

2. Two particles \(A\) and \(B\), of mass 0.3 kg and \(m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line so that the particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. In the collision the direction of motion of each particle is reversed and, immediately after the collision, the speed of each particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
2. the value of \(m\).

6. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 0.5 kg and \(m \mathrm {~kg}\) respectively, where \(m < 0.5\). The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially \(P\) is 3.15 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. After \(P\) has been descending for 1.5 s , it strikes the ground. Particle \(P\) reaches the ground before \(Q\) has reached the pulley.

1. Show that the acceleration of \(P\) as it descends is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string as \(P\) descends.
3. Show that \(m = \frac { 5 } { 18 }\).
4. State how you have used the information that the string is inextensible. When \(P\) strikes the ground, \(P\) does not rebound and the string becomes slack. Particle \(Q\) then moves freely under gravity, without reaching the pulley, until the string becomes taut again.
5. Find the time between the instant when \(P\) strikes the ground and the instant when the string becomes taut again.

5. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(A\) and \(B\) are above a horizontal plane, as shown in Figure 2. The system is released from rest.

1. Show that the tension in the string immediately after the particles are released is \(\frac { 12 } { 5 } m g\). After descending \(1.5 \mathrm {~m} , B\) strikes the plane and is immediately brought to rest. In the subsequent motion, \(A\) does not reach the pulley.
2. Find the distance travelled by \(A\) between the instant when \(B\) strikes the plane and the instant when the string next becomes taut. Given that \(m = 0.5 \mathrm {~kg}\),
3. find the magnitude of the impulse on \(B\) due to the impact with the plane.

1. Particle \(P\) of mass \(m\) and particle \(Q\) of mass \(k m\) are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of \(P\) is \(5 u\) and the speed of \(Q\) is \(u\). Immediately after the collision the speed of each particle is halved and the direction of motion of each particle is reversed.

Find

1. the value of \(k\),
2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.

3. A particle \(P\) of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, \(P\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between \(P\) and the ground is \(\frac { 1 } { 8 }\). Find the magnitude of the impulse exerted on \(P\) by the wall.

2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of \(P\) is \(4 u\) and the speed of \(Q\) is \(3 u\). As a result of the collision, \(Q\) has its direction of motion reversed and is moving with speed \(u\).

1. Find the speed of \(P\) immediately after the collision.
2. State whether or not the direction of motion of \(P\) has been reversed by the collision.
3. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.

4. A ball of mass 0.2 kg is projected vertically downwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 7 Ns in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

1. the value of \(U\). After hitting the ground, the ball moves vertically upwards and passes through a point \(B\) which is 1 m above the ground.
2. Find the time between the instant when the ball hits the ground and the instant when the ball first passes through \(B\).
3. Sketch a velocity-time graph for the motion of the ball from when it was projected from \(A\) to when it first passes through \(B\). (You need not make any further calculations to draw this sketch.)

1. A particle \(P\) of mass 0.6 kg is moving with velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { I } \mathrm { N }\) s. Immediately after receiving the impulse, \(P\) has velocity ( \(2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find

1. the magnitude of \(\mathbf { I }\),
2. the kinetic energy lost by \(P\) as a result of receiving the impulse.

4. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 20 \mathbf { i } - 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse ( \(- 6 \mathbf { i } + 8 \mathbf { j }\) ) N s.

1. Find the speed of \(P\) immediately after it receives the impulse.
   (5)
2. Find the size of the angle between the direction of motion of \(P\) before the impulse is received and the direction of motion of \(P\) after the impulse is received.
   (4)

1. A particle of mass 0.75 kg is moving with velocity ( \(4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse ( \(- 6 \mathbf { i } + 4 \mathbf { j }\) ) N s. impulse \(( - 6 \mathbf { i } + 4 \mathbf { j } )\) N s.

\section*{Find
Find} $$\begin{aligned} & \text { (a) the velocity of the particle immediately after receiving the impulse, } \\ & \text { (b) the size of the angle through which the path of the particle is deflected as a result of } \\ & \text { the impulse. } \end{aligned}$$ (3)

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.75 kg is moving along a straight line on a horizontal surface. At the instant when the speed of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it receives an impulse of magnitude \(\sqrt { 24 } \mathrm { Ns }\). The impulse acts in the plane of the horizontal surface. At the instant when \(P\) receives the impulse, the line of action of the impulse makes an angle of \(60 ^ { \circ }\) with the direction of motion of \(P\), as shown in Figure 2. Find

1. the speed of \(P\) immediately after receiving the impulse,
2. the size of the angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse. \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-06_2252_51_311_1980} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-07_36_65_2722_109}

1. A particle \(A\) has mass \(4 m\) and a particle \(B\) has mass \(3 m\). The particles 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 \(A\) is \(2 u\) and the speed of \(B\) is \(3 u\).
The direction of motion of each particle is reversed by the collision.
The total kinetic energy lost in the collision is \(\frac { 473 } { 24 } m u ^ { 2 }\) Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the magnitude of the impulse received by \(A\) in the collision.