6.03e Impulse: by a force

2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of \(P\) is \(4 u\) and the speed of \(Q\) is \(3 u\). As a result of the collision, \(Q\) has its direction of motion reversed and is moving with speed \(u\).

1. Find the speed of \(P\) immediately after the collision.
2. State whether or not the direction of motion of \(P\) has been reversed by the collision.
3. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.

4. A ball of mass 0.2 kg is projected vertically downwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 7 Ns in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

1. the value of \(U\). After hitting the ground, the ball moves vertically upwards and passes through a point \(B\) which is 1 m above the ground.
2. Find the time between the instant when the ball hits the ground and the instant when the ball first passes through \(B\).
3. Sketch a velocity-time graph for the motion of the ball from when it was projected from \(A\) to when it first passes through \(B\). (You need not make any further calculations to draw this sketch.)

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. A particle \(A\) has mass \(4 m\) and a particle \(B\) has mass \(3 m\). The particles are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(3 u\).
The direction of motion of each particle is reversed by the collision.
The total kinetic energy lost in the collision is \(\frac { 473 } { 24 } m u ^ { 2 }\) Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the magnitude of the impulse received by \(A\) in the collision.

1. A particle \(P\) of mass 1.5 kg is moving with velocity \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of magnitude 15Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(\boldsymbol { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the two possible values of \(v\).

8. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately after the collision, \(A\) and \(B\) are moving in the same direction with speeds \(\frac { u } { 3 }\) and \(u\) respectively. In the collision, \(A\) receives an impulse of magnitude 8mu.

1. Find the coefficient of restitution between \(A\) and \(B\). When \(A\) and \(B\) collide they are at a distance \(d\) from a smooth vertical wall, which is perpendicular to their direction of motion. After the collision with \(A\), particle \(B\) collides directly with the wall and rebounds so that there is a second collision between \(A\) and \(B\). This second collision takes place at distance \(x\) from the wall. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 4 }\)
2. find \(x\) in terms of \(d\).
   

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1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) The particle receives an impulse \(( - 2 \mathbf { i } + \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after receiving the impulse, the velocity of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The kinetic energy gained by \(P\) as a result of receiving the impulse is 22 J .

Find the possible values of \(\lambda\).

1. Particle \(P\) has mass \(3 m\) and particle \(Q\) has mass \(k m\). The particles are moving towards each other on the same straight line on a smooth horizontal surface.
   The particles collide directly.
   Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(v\).

The direction of motion of \(P\) is unchanged by the collision.

1. Show that \(v = \frac { ( 3 - 3 k ) } { k } u\)
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   Given that \(v \neq u\)
3. find the range of possible values of \(k\).

3. Two particles \(A\) and \(B\) have mass \(2 m\) and \(k m\) respectively. The particles are moving in opposite directions along the same straight smooth horizontal line so that the particles collide directly. Immediately before the collision \(A\) has speed \(2 u\) and \(B\) has speed \(u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { u } { 2 }\).

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(B\) on \(A\) in the collision.
2. Show that \(k < 5\)

2. A particle \(P\) has mass \(k m\) and a particle \(Q\) has mass \(m\). The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, \(P\) has speed \(3 u\) and \(Q\) has speed \(u\).
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 \(k\).
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(Q\) in the collision.

2. Two particles, \(A\) and \(B\), are moving in a straight line in opposite directions towards each other on a smooth horizontal surface when they collide directly. Particle \(A\) has mass \(3 m \mathrm {~kg}\) and particle \(B\) has mass \(m \mathrm {~kg}\).
Immediately before the collision, both particles have a speed of \(1.5 \mathrm {~ms} ^ { - 1 }\) Immediately after the collision, the direction of motion of \(A\) is unchanged and the difference between the speed of \(A\) and speed of \(B\) is \(1 \mathrm {~ms} ^ { - 1 }\)

1. Find (i) the speed of \(A\) immediately after the collision,
   (ii) the speed of \(B\) immediately after the collision.
2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) in the collision.

2. \begin{figure}[h] \end{figure} Figure 2 shows two particles, \(A\) and \(B\), moving in opposite directions on a smooth horizontal surface. Particle \(A\) has mass 5 kg and particle \(B\) has mass \(x \mathrm {~kg}\). The particles collide directly.
Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~ms} ^ { - 1 }\) and the speed of \(B\) is \(x \mathrm {~ms} ^ { - 1 }\) Immediately after the collision, the speed of \(A\) is \(1 \mathrm {~ms} ^ { - 1 }\) and its direction of motion is unchanged. Immediately after the collision, the speed of \(B\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the value of \(x\).
2. Find the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.

1. Two small smooth balls \(A\) and \(B\) have mass 0.6 kg and 0.9 kg respectively. They are moving in a straight line towards each other in opposite directions on a smooth horizontal floor and collide directly. Immediately before the collision the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after the collision and \(B\) is brought to rest by the collision.

Find

1. the value of \(v\),
2. the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.

2. Particle \(A\) of mass \(2 m\) and particle \(B\) of mass \(k m\), where \(k\) is a positive constant, are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision the speed of \(A\) is \(u\) and the speed of \(B\) is \(3 u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { 1 } { 2 } u\).

1. Show that \(k < 1\)
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by \(A\) in the collision.

3. Two particles, \(P\) and \(Q\), have masses 0.5 kg and \(m \mathrm {~kg}\) 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 \(4 \mathrm {~ms} ^ { - 1 }\) and the speed of \(Q\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is 4.2 N s . As a result of the collision the direction of motion of \(P\) is reversed.

1. Find the speed of \(P\) immediately after the collision. The speed of \(Q\) immediately after the collision is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the two possible values of \(m\).

1. Two particles, \(P\) and \(Q\), are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision the speed of \(Q\) is \(2 u\). The mass of \(Q\) is \(3 m\) and the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision is \(4 m u\).

Find

1. the speed of \(Q\) immediately after the collision,
2. the direction of motion of \(Q\) immediately after the collision.

1. A particle \(A\) has mass 4 kg and a particle \(B\) has mass 2 kg .

The particles move towards each other in opposite directions along the same straight line on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(2 u \mathrm {~ms} ^ { - 1 }\) and the speed of \(B\) is \(3 u \mathrm {~ms} ^ { - 1 }\) Immediately after the collision, the speed of \(B\) is \(2 u \mathrm {~ms} ^ { - 1 }\) The direction of motion of \(B\) is reversed by the collision.

1. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
2. State the direction of motion of \(A\) immediately after the collision.
3. Find, in terms of \(u\), the magnitude of the impulse received by \(B\) in the collision. State the units of your answer. \section*{[In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]}

1. Two particles, \(A\) and \(B\), have masses \(m\) and \(3 m\) respectively. The particles are connected by a light inextensible string. Initially \(A\) and \(B\) are at rest on a smooth horizontal plane with the string slack.

Particle \(A\) is then projected along the plane away from \(B\) with speed \(U\).
Given that the common speed of the particles immediately after the string becomes taut is \(S\)

1. find \(S\) in terms of \(U\).
2. Find, in terms of \(m\) and \(U\), the magnitude of the impulse exerted on \(A\) immediately after the string becomes taut.