Collision with fixed wall

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\includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

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\includegraphics[max width=\textwidth, alt={}, center]{699490ab-a01a-46e2-aa7c-3fd48c962c0c-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

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\includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

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.
   • The magnitude of the impulse that the wall exerts on \(P\).
   • The angle between \(\mathbf { i }\) and the impulse that the wall exerts on \(P\).

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.
   • The magnitude of \(\mathbf { I }\)
   • The angle between I and i
   • Find the loss of kinetic energy of \(P\) as a result of the collision.

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.

5 A smooth sphere is moving on a smooth horizontal surface when it strikes a smooth vertical wall and rebounds. Immediately before the impact, the sphere is moving with speed \(4 \mathrm {~ms} ^ { - 1 }\) and the angle between the sphere's direction of motion and the wall is \(\alpha\). Immediately after the impact, the sphere is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) and the angle between the sphere's direction of motion and the wall is \(40 ^ { \circ }\). The coefficient of restitution between the sphere and the wall is \(\frac { 2 } { 3 }\).
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-14_529_250_831_909}

1. Show that \(\tan \alpha = \frac { 3 } { 2 } \tan 40 ^ { \circ }\).
2. Find the value of \(v\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-15_2484_1709_223_153}

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.

2 A particle \(P\) of mass 0.2 kg is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a horizontal smooth surface. The direction of motion of \(P\) immediately before impact makes an angle of \(27 ^ { \circ }\) with the surface. Given that the coefficient of restitution between the particle and the surface is 0.6 , find

1. the vertical component of the velocity of \(P\) immediately after impact,
2. the magnitude of the impulse exerted on \(P\).

1 A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The speed of the sphere immediately after it bounces is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and hence find the coefficient of restitution between the surface and the sphere.
2. State the direction of the impulse on the sphere and find its magnitude.

2 A particle of mass 0.3 kg is projected horizontally under gravity with velocity \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point 0.4 m above a smooth horizontal plane. The particle first hits the plane at point \(A\); it bounces and hits the plane a second time at point \(B\). The distance \(A B\) is 1 m . Calculate

1. the vertical component of the velocity of the particle when it arrives at \(A\), and the time taken for the particle to travel from \(A\) to \(B\),
2. the coefficient of restitution between the particle and the plane,
3. the impulse exerted by the plane on the particle at \(A\).

4. \begin{figure}[h] \end{figure} A small smooth ball \(B\), moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(2 \theta\), where \(0 ^ { \circ } < \theta < 45 ^ { \circ }\). Immediately after the collision the angle between the direction of motion of \(B\) and the wall is \(\theta\), as shown in Figure 1. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 8 }\), find the value of \(\tan \theta\).
(8)

7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane] A small smooth ball of mass \(m\) kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector \(2 \mathbf { i } + \mathbf { j }\). The velocity of the ball immediately before the impact is \(b \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(b\) is positive. The velocity of the ball immediately after the impact is \(a ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(a\) is positive.

1. Show that the impulse received by the ball when it strikes the wall is parallel to \(( - \mathbf { i } + 2 \mathbf { j } )\). Find
2. the coefficient of restitution between the ball and the wall,
3. the fraction of the kinetic energy of the ball that is lost due to the impact.

2. A small ball \(B\), moving on a smooth horizontal plane, collides with a fixed smooth vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(\alpha\). The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). The kinetic energy of \(B\) immediately after the collision is \(60 \%\) of its kinetic energy immediately before the collision. Find, in degrees, the size of angle \(\alpha\).

1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).

Find the speed of \(S\) immediately after the collision.

7 A sphere, of mass 0.2 kg , moving on a smooth horizontal surface, collides with a fixed wall. Before the collision the sphere moves with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the wall. After the collision the sphere moves with speed \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) to the wall. The velocities are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-08_303_762_735_625} The coefficient of restitution between the wall and the sphere is 0.7 7

1. Assume that the wall is smooth. 7
2. 1. Find the value of \(v\) Give your answer to two significant figures.
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3. (ii) Find the value of \(\theta\) Give your answer to the nearest whole number.
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4. (iii) Find the magnitude of the impulse exerted on the sphere by the wall.
   Give your answer to two significant figures.
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5. In reality the wall is not smooth.
   Explain how this would cause a change in the magnitude of the impulse calculated in part (a)(iii).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a smooth horizontal floor, where \(A B\) represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.
Immediately before the impact the velocity of the ball is \(( 7 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Immediately after the impact the velocity of the ball is \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
The coefficient of restitution between the ball and the wall is \(e\).

1. Show that \(A B\) is parallel to \(( 2 \mathbf { i } + 3 \mathbf { j } )\).
2. Find the value of \(e\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h] \end{figure} Figure 3 represents the plan view of part of a smooth horizontal floor, where \(A B\) is a fixed smooth vertical wall. The direction of \(\overrightarrow { A B }\) is in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\)
A small ball of mass 0.25 kg is moving on the floor when it strikes the wall \(A B\).
Immediately before its impact with the wall \(A B\), the velocity of the ball is \(( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
Immediately after its impact with the wall \(A B\), the velocity of the ball is \(\mathbf { v m s } ^ { - 1 }\)
The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\)
By modelling the ball as a particle,

1. show that \(\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }\)
2. Find the magnitude of the impulse received by the ball in the impact.

6 A particle \(P\) of mass 2.5 kg strikes a rough horizontal plane. Immediately before \(P\) strikes the plane it has a speed of \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion makes an angle of \(30 ^ { \circ }\) with the normal to the plane at the point of impact. The impact may be assumed to occur instantaneously. The coefficient of restitution between \(P\) and the plane is \(\frac { 2 } { 3 }\). The friction causes a horizontal impulse of magnitude 2 Ns to be applied to \(P\) in the plane in which it is moving.

1. Calculate the velocity of \(P\) immediately after the impact with the plane.
2. \(\quad P\) loses about \(x \%\) of its kinetic energy as a result of the impact. Find the value of \(x\).

6 A ball moving on a smooth horizontal surface collides with a fixed vertical wall. Before the collision, the ball moves with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at an angle of \(40 ^ { \circ }\) to the wall. After the collision, the ball moves with speed \(5 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(26 ^ { \circ }\) to the wall. Model the ball as a particle.
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1. Find the coefficient of restitution between the ball and the wall, giving your answer correct to two significant figures.
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2. Determine whether or not the wall is smooth. 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]