Angle change from 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. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(Q\) of mass 0.5 kg is moving on a smooth horizontal surface. Particle \(Q\) is moving with velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\).

1. Find the speed of \(Q\) immediately after receiving the impulse. As a result of receiving the impulse, the direction of motion of \(Q\) is turned through an angle \(\theta ^ { \circ }\)
2. Find the value of \(\theta\)

5. \begin{figure}[h] \end{figure} A small ball \(B\) of mass 0.25 kg is moving in a straight line with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it is given an impulse. The impulse has magnitude 12.5 N s and is applied in a horizontal direction making an angle of \(\left( 90 ^ { \circ } + \alpha \right)\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the initial direction of motion of the ball, as shown in Figure 3.

1. Find the speed of \(B\) immediately after the impulse is applied.
2. Find the direction of motion of \(B\) immediately after the impulse is applied.

1. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { Ns }\). The angle between the velocity of \(P\) before the impulse and the velocity of \(P\) after the impulse is \(\theta ^ { \circ }\).

Find

1. the value of \(\theta\),
2. the kinetic energy gained by \(P\) as a result of the impulse.

1 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-2_385_741_269_701} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse of magnitude 2.6 N s applied to \(P\) deflects its direction of motion through an angle \(\theta\), and reduces its speed to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). By considering an impulse-momentum triangle, or otherwise,

1. show that \(\cos \theta = 0.6\),
2. find the angle that the impulse makes with the original direction of motion of \(P\).

1 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_323_639_255_753} A particle \(P\) of mass 0.4 kg is moving horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\), in a direction which makes an angle \(( 180 - \theta ) ^ { \circ }\) with the direction of motion of \(P\). Immediately after the impulse acts \(P\) moves horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is turned through an angle of \(60 ^ { \circ }\) by the impulse (see diagram). Find \(I\) and \(\theta\).

1 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_476_583_258_781} A ball of mass 0.5 kg is moving with speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line when it is struck by a bat. The impulse exerted by the bat has magnitude 15 N s and the ball is deflected through an angle of \(90 ^ { \circ }\) (see diagram). Find

1. the direction of the impulse,
2. the speed of the ball immediately after it is struck.

1 \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_477_534_261_770} A ball of mass 0.6 kg is moving with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line. It is struck by an impulse \(I \mathrm { Ns }\) acting at an acute angle \(\theta\) to its direction of motion (see diagram). The impulse causes the direction of motion of the ball to change by an acute angle \(\alpha\), where \(\sin \alpha = \frac { 8 } { 17 }\). After the impulse acts the ball is moving with a speed of \(3.4 \mathrm {~ms} ^ { - 1 }\). Find \(I\) and \(\theta\).

1 A ball of mass 0.4 kg is moving in a straight line, with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is struck by a bat. The bat exerts an impulse of magnitude 20 N s and the ball is deflected through an angle of \(90 ^ { \circ }\). Calculate

1. the direction of the impulse,
2. the speed of the ball immediately after it is struck.

2 A tennis ball of mass 0.057 kg has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 0.6 N s which reduces the speed of the ball to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Using an impulse-momentum triangle, or otherwise, find the angle the impulse makes with the original direction of motion of the ball.

A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of \(P\) makes an angle \(\alpha\) with the barrier. The coefficient of restitution between \(P\) and the barrier is \(e\). As a result of the impact, the direction of motion of \(P\) is turned through \(90°\).

1. Show that \(\tan^2 \alpha = \frac{1}{e}\). [3]

The particle \(P\) loses two-thirds of its kinetic energy in the impact.

1. Find the value of \(\alpha\) and the value of \(e\). [5]

A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\frac{1}{3}\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision. Find the angle through which the path of the ball is deflected by the collision. [8]