Impulse from velocity change

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.

1. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse \(\mathbf { I N }\) s, such that \(\mathbf { I } = a \mathbf { i } + b \mathbf { j }\)

Immediately after receiving the impulse, the particle is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a constant. Given that the magnitude of \(\mathbf { I }\) is \(\sqrt { 40 }\), find the two possible impulses.
(5)

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.3 kg is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane. The particle receives a horizontal impulse of magnitude \(J\) Ns. The speed of \(P\) immediately after receiving the impulse is \(8 \mathrm {~ms} ^ { - 1 }\). The angle between the direction of motion of \(P\) before it receives the impulse and the direction of the impulse is \(60 ^ { \circ }\), as shown in Figure 2. Find the value of \(J\).
(6)

1. A particle \(P\) of mass 0.3 kg is moving with velocity \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The particle receives an impulse I Ns.
Immediately after receiving the impulse, the velocity of \(P\) is \(( 7 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the magnitude of \(\mathbf { I }\)
2. Find the angle between the direction of \(\mathbf { I }\) and the direction of motion of \(P\) immediately before receiving the impulse.

3. \begin{figure}[h] \end{figure} A particle \(Q\) of mass 0.25 kg is moving in a straight line on a smooth horizontal surface with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\). The impulse acts parallel to the horizontal surface and at \(60 ^ { \circ }\) to the original direction of motion of \(Q\). Immediately after receiving the impulse, the speed of \(Q\) is \(12 \mathrm {~ms} ^ { - 1 }\) As a result of receiving the impulse, the direction of motion of \(Q\) is turned through \(\alpha ^ { \circ }\), as shown in Figure 2. Find the value of \(I\)

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity \(( 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the impulse of the bat on the ball,
2. the size of the angle between the vector \(\mathbf { i }\) and the impulse exerted by the bat on the ball,
3. the kinetic energy lost by the ball in the impact.

1. A ball of mass 0.4 kg is moving in a horizontal plane when it is struck by a bat. The bat exerts an impulse \(( - 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) s on the ball. Immediately after receiving the impulse the ball has velocity \(( 12 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find

1. the speed of the ball immediately before the impact,
2. the size of the angle through which the direction of motion of the ball is deflected by the impact.

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse I Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

Find

1. the magnitude of \(\mathbf { I }\),
2. the angle between \(\mathbf { I }\) and \(\mathbf { j }\).

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) lie on a smooth horizontal plane. A small ball of mass 0.2 kg is moving along the line \(A B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the ball is at \(B\), the ball is given an impulse. Immediately after the impulse is given, the ball moves along the line \(B C\) with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The line \(B C\) makes an angle of \(35 ^ { \circ }\) with the line \(A B\), as shown in Figure 1.

1. Find the magnitude of the impulse given to the ball.
2. Find the size of the angle between the direction of the impulse and the original direction of motion of the ball.

1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find

1. the magnitude of the impulse that acts on the bullet,
2. the thickness of the barrier,
3. the time taken for the bullet to pass through the barrier.

1. A ball of mass 0.6 kg bounces against a wall and is given an impulse of \(( 12 \mathbf { i } - 9 \mathbf { j } ) \mathrm { Ns }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors. The velocity of the particle after the impact is \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the velocity of the particle before the impact.
(4 marks)

1. An ice hockey puck of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity \(( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)

1 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle \(P\) of mass 0.4 kg moving in a straight line has speed \(8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse applied to \(P\) deflects it through \(45 ^ { \circ }\) and reduces its speed to \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Calculate the magnitude and direction of the impulse exerted on \(P\). \(2 \quad O\) is a fixed point on a horizontal straight line. A particle \(P\) of mass 0.5 kg is released from rest at \(O\). At time \(t\) seconds after release the only force acting on \(P\) has magnitude \(\left( 1 + k t ^ { 2 } \right) \mathrm { N }\) and acts horizontally and away from \(O\) along the line, where \(k\) is a positive constant.

1. Find the speed of \(P\) in terms of \(k\) and \(t\).
2. Given that \(P\) is 2 m from \(O\) when \(t = 1\), find the value of \(k\) and the time taken by \(P\) to travel 20 m from \(O\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_231_971_539_587} When a tennis ball of mass 0.057 kg bounces it receives an impulse of magnitude \(I \mathrm {~N} \mathrm {~s}\) at an angle of \(\theta\) to the horizontal. Immediately before the ball bounces it has speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal. Immediately after the ball bounces it has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal (see diagram). Find \(I\) and \(\theta\).

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

1. Find the magnitude of the impulse. [5]
2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied. [3]

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]