6.03f Impulse-momentum: relation

6 At a fairground a dodgem car is moving in a straight horizontal line towards a side wall that is perpendicular to the velocity of the car. The speed of the car is \(1.8 \mathrm {~ms} ^ { - 1 }\) It collides with the side wall and rebounds along its original path with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total mass of the dodgem car and the passengers is 250 kg
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1. Find the magnitude of the impulse on the car during the collision with the side wall.
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2. A possible model for the magnitude of the force, \(F\) newtons, acting on the dodgem car due to its collision with the side wall is given by $$F = k t ( 4 - 5 t ) \quad \text { for } 0 \leq t \leq 0.8$$ 6 (b) (i) Find the value of \(k\).
   (b) (ii) Determine the maximum magnitude of the force predicted by the model. 6 (b) (ii) Determine the maximum magnitude of the fored bed bed at

6 An ice hockey puck, of mass 0.2 kg , is moving in a straight line on a horizontal ice rink under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude ( \(2 t + 3\) ) newtons.
The force acts on the puck from \(t = 0\) to \(t = T\) 6

1. Show that the magnitude of the impulse of the force is \(a T ^ { 2 } + b T\), where \(a\) and \(b\) are integers to be found.
   [0pt] [3 marks]
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2. While the force acts on the puck, its speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Use your answer from part (a) to find \(T\), giving your answer to three significant figures.
   Fully justify your answer.

5 A ball, of mass 0.3 kg , is moving on a smooth horizontal surface. The ball collides with a smooth fixed vertical wall and rebounds.
Before the ball hits the wall, the ball is moving at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the wall as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-06_634_268_584_886} The magnitude of the force, \(F\) newtons, exerted on the ball by the wall at time \(t\) seconds is modelled by $$F = k t ^ { 2 } ( 0.1 - t ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.1$$ where \(k\) is a constant. The ball is in contact with the wall for 0.1 seconds. 5 (b) Explain why \(1800000 < k \leq 3600000\) Fully justify your answer.
5 (c) Given that \(k = 2400000\) Find the speed of the ball after the collision with the wall.
[0pt] [4 marks]

1 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.

1 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.

2 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 . \begin{enumerate}[label=(\alph*)] \item 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 \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\). \item Find the value of \(R\). \item 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 . \item

1. Find the speed of \(Q\) after the collision.
2. Hence show that the collision is inelastic. It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274} In a model, the following modelling assumptions are made.
   For a particular portion of banked track, \(r = 24\).
   (b) Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
   (c) Explain

4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).

1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(I = \frac { \sqrt { 3 } m v } { 2 ( 1 + m ) }\).
2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.

1. Particle \(A\) has mass \(4 m\) and particle \(B\) has mass \(3 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, the speed of \(A\) is \(2 x\) and the speed of \(B\) is \(x\).
Immediately after the collision, the speed of \(A\) is \(y\) and the speed of \(B\) is \(5 y\).
The direction of motion of each particle is reversed as a result of the collision.

1. Show that \(y = \frac { 5 } { 11 } x\).
2. Find, in terms of \(m\) and \(x\), the magnitude of the impulse received by \(A\) in the collision.

7 At time \(t = 0\), a small ball \(A\) is projected vertically upwards with speed \(8 \mathrm {~ms} ^ { - 1 }\) from a fixed point on horizontal ground.
The ball hits the ground again for the first time at time \(t = T _ { 1 }\) seconds.
Ball \(A\) is modelled as a particle moving freely under gravity.

1. Show that \(T _ { 1 } = 1.63\) to 3 significant figures. After the first impact with the ground, \(A\) rebounds to a height of 2 m above the ground.
   Given that the mass of \(A\) is 0.1 kg ,
2. find the magnitude of the impulse received by \(A\) as a result of its first impact with the ground. At time \(t = 1\) second, another small ball \(B\) is projected vertically upwards from another point on the ground with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Ball \(B\) is modelled as a particle moving freely under gravity.
   At time \(t = T _ { 2 }\) seconds ( \(T _ { 2 } > 1\) ), \(A\) and \(B\) are at the same height above the ground for the first time.
3. Find the value of \(T _ { 2 }\)

1. A particle \(Q\) of mass 3 kg is moving on a smooth horizontal surface.

Particle \(Q\) is moving with velocity \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives a horizontal impulse of magnitude \(3 \sqrt { 82 } \mathrm { Ns }\). Immediately after receiving the impulse, the velocity of \(Q\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(x\) and \(y\) are positive constants. The kinetic energy gained by \(Q\) as a result of receiving the impulse is 138 J .
Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(Q\) immediately after receiving the impulse.

11 A smooth sphere of mass 2 kg has velocity \(( 24 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and is travelling on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. The sphere strikes a vertical wall. The line of intersection of the wall and the plane is in the direction \(( 4 \mathbf { i } + 3 \mathbf { j } )\).

1. Show that the acute angle between the path of the sphere before the impact and the direction of the wall is \(\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)\).
2. After the impact, the velocity of the sphere is \(( 7.2 \mathbf { i } + 15.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
   1. the coefficient of restitution between the sphere and the wall,
   2. the magnitude of the impulse exerted by the sphere on the wall.

11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane \(\Pi\) that makes an angle of \(\sin ^ { - 1 } 0.6\) above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of \(\Pi\). Immediately before the first impact between the particle and \(\Pi\), the particle is moving horizontally with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the particle and \(\Pi\) is 0.5 .

1. Find the magnitude of the impulse on the particle from \(\Pi\) at the first impact, and state the direction of this impulse.
2. Find the distance between the points on \(\Pi\) where the first and second impacts occur.
3. Find the time taken between the first and third impacts.

Two particles, of masses \(3m\) and \(m\), are moving in the same straight line towards each other with speeds \(2u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4mu\). Show that the total loss in kinetic energy is \(\frac{4}{5}mu^2\). [6]

Two particles, of masses \(3m\) and \(m\), are moving in the same straight line towards each other with speeds \(2u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4mu\). Show that the total loss in kinetic energy is \(\frac{5}{2}mu^2\). [6]

A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed 300 m s\(^{-1}\) and emerges horizontally after 0.02 s. There is a constant horizontal resisting force of magnitude 1000 N. Find the speed with which the bullet emerges from the barrier. [3]

\includegraphics{figure_1} A uniform disc with centre \(O\), mass \(m\) and radius \(a\) is free to rotate without resistance in a vertical plane about a horizontal axis through \(O\). One end of a light inextensible string is attached to the rim of the disc and wrapped around the rim. The other end of the string is attached to a block of mass \(3m\) (see diagram). The system is released from rest with the block hanging vertically. While the block is in motion, it experiences a constant vertical resisting force of magnitude \(0.9mg\). Find the tension in the string in terms of \(m\) and \(g\). [5]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\). Show that the least possible value of \(u\) is \(\sqrt{(ag)}\). [2] It is now given that \(u = \sqrt{(ag)}\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac{1}{4}m\). Find the speed of the combined particle immediately after the collision. [4] In the subsequent motion, when \(OP\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\). [5] Find the value of \(\cos \theta\) when the string becomes slack. [2]

A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3u\) and the speed of truck \(B\) is \(2u\). In the collision the trucks join together. Modelling the trucks as particles, find

1. the speed of \(A\) immediately after the collision, [3]
2. the direction of motion of \(A\) immediately after the collision, [1]
3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision. [3]

Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) is moving due east and particle \(Q\) is moving due west. Particle \(P\) has mass \(2m\) and particle \(Q\) has mass \(3m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(4u\) and the speed of \(Q\) is \(u\). The magnitude of the impulse in the collision is \(\frac{33}{5}mu\).

1. Find the speed and direction of motion of \(P\) immediately after the collision. [4]
2. Find the speed and direction of motion of \(Q\) immediately after the collision. [4]

Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(km\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2u\). As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.

1. Find the value of \(k\). [4]
2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision. [2]

Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(4u\). In the collision, the particles join together to form a single particle. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision. [6]

A railway truck \(S\) of mass 20 tonnes is moving along a straight horizontal track when it collides with another railway truck \(T\) of mass 30 tonnes which is at rest. Immediately before the collision the speed of \(S\) is \(4\text{ ms}^{-1}\) As a result of the collision, the two railway trucks join together. Find

1. the common speed of the railway trucks immediately after the collision, [2]
2. the magnitude of the impulse exerted on \(S\) in the collision, stating the units of your answer. [3]

Particle \(P\) has mass \(m\) kg and particle \(Q\) has mass \(3m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4u\) m s\(^{-1}\) and \(Q\) has speed \(ku\) m s\(^{-1}\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.

1. Find the value of \(k\). [4]
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\). [3]

A ball of mass 0.3 kg is moving vertically downwards with speed 8 m s\(^{-1}\) when it hits the floor which is smooth and horizontal. It rebounds vertically from the floor with speed 6 m s\(^{-1}\). Find the magnitude of the impulse exerted by the floor on the ball. [3]

A railway truck \(P\) of mass 2000 kg is moving along a straight horizontal track with speed 10 m s\(^{-1}\). The truck \(P\) collides with a truck \(Q\) of mass 3000 kg, which is at rest on the same track. Immediately after the collision \(Q\) moves with speed 5 m s\(^{-1}\). Calculate

1. the speed of \(P\) immediately after the collision, [3]
2. the magnitude of the impulse exerted by \(P\) on \(Q\) during the collision. [2]