6.03e Impulse: by a force

5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\).
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, stating the units in which your answer is given.
   (3 marks)
   The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
3. Find the coefficient of friction between \(B\) and the table.
   (6 marks)

2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).

1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Hence find the magnitude of the frictional force acting on \(P\). \(P\) first comes to instantaneous rest at point \(C\) on the plane.
4. Calculate \(A C\).

2 A particle \(A\) of mass 3.6 kg is attached by a light inextensible string to a particle \(B\) of mass 2.4 kg . \(A\) and \(B\) are initially at rest, with the string slack, on a smooth horizontal surface. \(A\) is projected directly away from \(B\) with a speed of \(7.2 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the speed of \(A\) after the string becomes taut.
2. Find the impulse exerted on \(A\) at the instant that the string becomes taut.
3. Find the loss in kinetic energy as a result of the string becoming taut.

2 A hockey puck of mass 0.2 kg is sliding down a rough slope which is inclined at \(10 ^ { \circ }\) to the horizontal. At the instant that its velocity is \(14 \mathrm {~ms} ^ { - 1 }\) directly down the slope it is hit by a hockey stick. Immediately after it is hit its velocity is \(24 \mathrm {~ms} ^ { - 1 }\) directly up the slope.

1. Find the magnitude of the impulse exerted by the hockey stick on the puck. After it has been hit, the puck first comes to instantaneous rest when it has travelled 15 m up the slope. While the puck is moving up the slope, the resistance to its motion has constant magnitude \(R \mathrm {~N}\).
2. Use an energy method to determine the value of \(R\).

1 Two particles \(A\), of mass \(m \mathrm {~kg}\), and \(B\), of mass \(3 m \mathrm {~kg}\), are connected by a light inextensible string and placed together at rest on a smooth horizontal surface with the string slack. \(A\) is projected along the surface, directly away from \(B\), with a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(B\) immediately after the string becomes taut.
2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) as a result of the string becoming taut.

3 A crate of mass 45 kg is sliding with a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) in a straight line across a smooth horizontal floor. One end of a light inextensible rope is attached to the crate. At a certain instant a builder takes the other end of the rope and starts to pull, applying a constant force of 80 N for 5 seconds. While the builder is pulling the crate, the rope makes a constant angle of \(40 ^ { \circ }\) above the horizontal. Both the rope and the velocity of the crate lie in the same vertical plane (see diagram).
It may be assumed that there is no resistance to the motion of the crate.

1. Determine the work done by the builder in pulling the crate.
2. 1. Find the kinetic energy of the crate at the instant when the builder stops pulling the crate.
   2. Explain why the answers to part (a) and part (b)(i) are not equal.
3. Find the average power developed by the builder in pulling the crate.
4. Calculate the total impulse exerted on the crate by the builder.

1 A particle \(P\) of mass 2.5 kg is moving with a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal plane when it collides directly with a fixed vertical wall. After the collision \(P\) moves away from the wall with a speed of \(2.8 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the coefficient of restitution between \(P\) and the wall.
2. Find the magnitude and state the direction of the impulse exerted on \(P\) by the wall.
3. State the magnitude and direction of the impulse exerted on the wall by \(P\).

7 A particle \(P\) of mass 3.5 kg is attached to one end of a rod of length 5.4 m . The other end of the rod is hinged at a fixed point \(O\) and \(P\) hangs in equilibrium directly below \(O\). A horizontal impulse of magnitude 44.1 Ns is applied to \(P\).
In an initial model of the subsequent motion of \(P\) the rod is modelled as being light and inextensible and all resistance to the motion of \(P\) is ignored. You are given that \(P\) moves in a circular path in a vertical plane containing \(O\). The angle that the rod makes with the downward vertical through \(O\) is \(\theta\) radians.

1. Determine the largest value of \(\theta\) in the subsequent motion of \(P\). In a revised model the rod is still modelled as being light and inextensible but the resistance to the motion of \(P\) is not ignored. Instead, it is modelled as causing a loss of energy of 20 J for every metre that \(P\) travels.
2. Show that according to the revised model, the maximum value of \(\theta\) in the subsequent motion of \(P\) satisfies the following equation. $$343 ( 1 + 2 \cos \theta ) = 400 \theta$$ You are given that \(\theta = 1.306\) is the solution to the above equation, correct to \(\mathbf { 4 }\) significant figures.
3. Determine the difference in the predicted maximum vertical heights attained by \(P\) using the two models. Give your answer correct to \(\mathbf { 3 }\) significant figures.
4. Suggest one further improvement that could be made to the model of the motion of \(P\).

2 A particle \(P\) of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\) when it strikes a vertical wall directly. It rebounds at a speed of \(1.6 \mathrm {~ms} ^ { - 1 }\).

1. Find the coefficient of restitution between \(P\) and the wall.
2. Determine the impulse applied to \(P\) by the wall, stating its direction.
3. Find the loss of kinetic energy of \(P\) as a result of the collision.
4. State, with a reason, whether the collision is perfectly elastic.

4 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m .

1. Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~ms} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\).
2. Find the value of \(R\).
3. Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns .
4. 1. Find the speed of \(Q\) after the collision.
   2. Hence show that the collision is inelastic.

3 A particle \(A\) of mass 0.5 kg is moving with a speed of \(3.15 \mathrm {~ms} ^ { - 1 }\) on a smooth horizontal surface when it collides directly with a particle \(B\) of mass 0.8 kg which is at rest on the surface. The velocities of \(A\) and \(B\) immediately after the collision are denoted by \(\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and \(\mathrm { v } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. You are given that \(\mathrm { v } _ { \mathrm { B } } = 2 \mathrm { v } _ { \mathrm { A } }\).

1. Find the values of \(\mathrm { V } _ { \mathrm { A } }\) and \(\mathrm { V } _ { \mathrm { B } }\).
2. Find the coefficient of restitution between \(A\) and \(B\).
3. Explain why the coefficient of restitution is a dimensionless quantity.
4. Calculate the total loss of kinetic energy as a result of the collision.
5. State, giving a reason, whether or not the collision is perfectly elastic.
6. Calculate the impulse that \(B\) exerts on \(A\) in the collision.

1 A particle \(P\) of mass 12.5 kg is moving on a smooth horizontal plane when it collides obliquely with a fixed vertical wall. At the instant before the collision, the velocity of \(P\) is \(- 5 \mathbf { i } + 12 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
At the instant after the collision, the velocity of \(P\) is \(\mathbf { i } + 4 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).

1. Find the magnitude of the momentum of \(P\) before the collision.
2. Find, in vector form, the impulse that the wall exerts on \(P\).
3. State, in vector form, the impulse that \(P\) exerts on the wall.
4. Find in either order.

2 A particle \(P\) of mass 2 kg is moving on a large smooth horizontal plane when it collides with a fixed smooth vertical wall. Before the collision its velocity is \(( 5 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision its velocity is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. The impulse imparted on \(P\) by the wall is denoted by INs. Find the following.

1 When a stationary firework P of mass 0.4 kg is set off, the explosion gives it an instantaneous impulse of 16 N s vertically upwards.

1. Calculate the speed of projection of P . While travelling vertically upwards at \(32 \mathrm {~ms} ^ { - 1 } , P\) collides directly with another firework \(Q\), of mass 0.6 kg , that is moving directly downwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1. The coefficient of restitution in the collision is 0.1 and Q has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards immediately after the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-2_520_422_753_817} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
2. Show that \(u = 18\) and calculate the speed and direction of motion of P immediately after the collision. Another firework of mass 0.5 kg has a velocity of \(( - 3.6 \mathbf { i } + 5.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors, respectively. This firework explodes into two parts, C and D . Part C has mass 0.2 kg and velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) immediately after the explosion.
3. Calculate the velocity of D immediately after the explosion in the form \(a \mathbf { i } + b \mathbf { j }\). Show that C and D move off at \(90 ^ { \circ }\) to one another.
   [0pt] [8]

3 A ball of mass 0.45 kg is travelling horizontally with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time \(t\) seconds after first coming into contact with the bat, the force exerted on the ball by the bat is \(1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)\) newtons, where \(0 \leqslant t \leqslant 0.1\). In this simple model, ignore the weight of the ball and model the ball as a particle.

1. Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
2. Find, to two significant figures, the speed of the ball immediately after the impact.
3. Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.

3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).

1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
2. Find the speed of \(P\) when \(t = 3\).
3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).

3 A particle of mass 0.2 kg lies at rest on a smooth horizontal table. A horizontal force of magnitude \(F\) newtons acts on the particle in a constant direction for 0.1 seconds. At time \(t\) seconds, $$F = 5 \times 10 ^ { 3 } t ^ { 2 } , \quad 0 \leqslant t \leqslant 0.1$$ Find the value of \(t\) when the speed of the particle is \(2 \mathrm {~ms} ^ { - 1 }\).
(4 marks)

4 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(m\) and \(2 m\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) when it collides with the sphere \(B\) which has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } )\). After the collision, the velocity of \(B\) is \(( \mathbf { i } + \mathbf { j } )\).

1. Find the velocity of \(A\) immediately after the collision.
2. Find the angle between the velocities of \(A\) and \(B\) immediately after the collision.
3. Find the impulse exerted by \(B\) on \(A\).
4. State, as a vector, the direction of the line of centres of \(A\) and \(B\) when they collide.
   (1 mark)

6 A small smooth ball of mass \(m\), moving on a smooth horizontal surface, hits a smooth vertical wall and rebounds. The coefficient of restitution between the wall and the ball is \(\frac { 3 } { 4 }\). Immediately before the collision, the ball has velocity \(u\) and the angle between the ball's direction of motion and the wall is \(\alpha\). The ball's direction of motion immediately after the collision is at right angles to its direction of motion before the collision, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-4_483_344_657_854}

1. Show that \(\tan \alpha = \frac { 2 } { \sqrt { 3 } }\).
2. Find, in terms of \(u\), the speed of the ball immediately after the collision.
3. The force exerted on the ball by the wall acts for 0.1 seconds. Given that \(m = 0.2 \mathrm {~kg}\) and \(u = 4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the average force exerted by the wall on the ball.

3 Three smooth spheres, \(A , B\) and \(C\), of equal radii have masses \(1 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(x \mathrm {~kg}\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(3 u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is \(\frac { 1 } { 3 }\).

1. Show that \(A\) is brought to rest by the impact and find the speed of \(B\) immediately after the collision in terms of \(u\).
2. Subsequently, \(B\) collides with \(C\). Show that the speed of \(C\) immediately after the collision is \(\frac { 4 u } { 3 + x }\).
   Find the speed of \(B\) immediately after the collision in terms of \(u\) and \(x\).
3. Show that \(B\) will collide with \(A\) again if \(x > 9\).
4. Given that \(x = 5\), find the magnitude of the impulse exerted on \(C\) by \(B\) in terms of \(u\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}

1 A ball of mass 0.2 kg is hit directly by a bat. Just before the impact, the ball is travelling horizontally with speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Just after the impact, the ball is travelling horizontally with speed \(32 \mathrm {~ms} ^ { - 1 }\) in the opposite direction.

1. Find the magnitude of the impulse exerted on the ball.
2. At time \(t\) seconds after the ball first comes into contact with the bat, the force exerted by the bat on the ball is \(k \left( 0.9 t - 10 t ^ { 2 } \right)\) newtons, where \(k\) is a constant and \(0 \leqslant t \leqslant 0.09\). The bat stays in contact with the ball for 0.09 seconds. Find the value of \(k\).

1 An ice-hockey player has mass 60 kg . He slides in a straight line at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on the horizontal smooth surface of an ice rink towards the vertical perimeter wall of the rink, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-02_476_594_769_715} The player collides directly with the wall, and remains in contact with the wall for 0.5 seconds. At time \(t\) seconds after coming into contact with the wall, the force exerted by the wall on the player is \(4 \times 10 ^ { 4 } t ^ { 2 } ( 1 - 2 t )\) newtons, where \(0 \leqslant t \leqslant 0.5\).

1. Find the magnitude of the impulse exerted by the wall on the player.
2. The player rebounds from the wall. Find the player's speed immediately after the collision.

4 The diagram shows part of a horizontal snooker table of width 1.69 m . A player strikes the ball \(B\) directly, and it moves in a straight line. The ball hits the cushion of the table at \(C\) before rebounding and moving to the pocket at \(P\) at the corner of the table, as shown in the diagram. The point \(C\) is 1.20 m from the corner \(A\) of the table. The ball has mass 0.15 kg and, immediately before the collision with the cushion, it has velocity \(u\) in a direction inclined at \(60 ^ { \circ }\) to the cushion. The table and the cushion are modelled as smooth. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-08_517_963_719_511}

1. Find the coefficient of restitution between the ball and the cushion.
2. Show that the magnitude of the impulse on the cushion at \(C\) is approximately \(0.236 u\).
3. Find, in terms of \(u\), the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\).
4. Explain how you have used the assumption that the cushion is smooth in your answers.

7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it collides with the sphere \(B\), which has velocity \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision, the velocity of the sphere \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Show that the impulse exerted on \(B\) in the collision is \(( 6 m \mathbf { j } )\) Ns.
3. Find the coefficient of restitution between the two spheres.
4. After the collision, each sphere moves in a straight line with constant speed. Given that the radius of each sphere is 0.05 m , find the time taken, from the collision, until the centres of the spheres are 1.10 m apart.

1 A stone, of mass 2 kg , is moving in a straight line on a smooth horizontal sheet of ice under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude \(( 3 t + 1 )\) newtons, \(0 \leqslant t \leqslant 3\). When \(t = 0\), the stone has velocity \(1 \mathrm {~ms} ^ { - 1 }\).
When \(t = T\), the stone has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(T\).
(6 marks)