6.03f Impulse-momentum: relation

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 \(\lambda\).

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 \(k < 5\)

1. Two particles, \(P\) and \(Q\), of mass \(m _ { 1 }\) and \(m _ { 2 }\) respectively, are moving on a smooth horizontal plane. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, both particles are moving with speed \(u\).

The direction of motion of each particle is reversed by the collision.
Immediately after the collision, the speed of \(Q\) is \(\frac { 1 } { 3 } u\).

1. Find, in terms of \(m _ { 2 }\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision.
2. Find, in terms of \(m _ { 1 } , m _ { 2 }\) and \(u\), the speed of \(P\) immediately after the collision.
3. Hence show that \(m _ { 2 } > \frac { 3 } { 4 } m _ { 1 }\)

2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(m\) respectively. The particles 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 \(P\) is \(3 u\) and the speed of \(Q\) is \(2 u\). The magnitude of the impulse exerted on \(Q\) by \(P\) in the collision is 5mu. Find

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

1. A particle \(P\) has mass \(3 m\) and a particle \(Q\) has mass \(5 m\). The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision the speed of \(P\) is \(k u\), where \(k\) is a constant, and the speed of \(Q\) is \(2 u\). Immediately after the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(3 u\).
The direction of motion of \(Q\) is reversed by the collision.

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
2. Find the two possible values of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-03_2647_1837_118_114}

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.

1. A hammer is used to hit a tent peg into soft ground.

The hammer has mass 1.8 kg and the tent peg has mass 0.2 kg .
The hammer and tent peg are both modelled as particles and the impact is modelled as a direct collision. Immediately before the impact, the tent peg is stationary and the hammer is moving vertically downwards with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Immediately after the impact, the hammer and tent peg move together, vertically downwards, with the same speed \(v \mathrm {~ms} ^ { - 1 }\)

1. Find the value of \(v\)
2. Find the magnitude of the impulse exerted on the tent peg by the hammer, stating the units of your answer. The ground exerts a constant vertical resistive force of magnitude \(R\) newtons, bringing the hammer and tent peg to rest after they travel a distance of 12 cm .
3. Find the value of \(R\).

2. 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 \(k m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\). 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 \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.

5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\). By modelling the balls as particles, find

1. the speed of \(B\) immediately after the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision, stating the units in which your answer is given. The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
3. Find the coefficient of friction between \(B\) and the table.

3. Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(k m\), where \(2 < k < 3\), and the mass of \(B\) is \(m\). The particles are moving along the same straight line, but in opposite directions, and they collide directly. Immediately before they collide the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(4 u\). As a result of the collision the speed of \(A\) is halved and its direction of motion is reversed.

1. Find, in terms of \(k\) and \(u\), the speed of \(B\) immediately after the collision.
2. State whether the direction of motion of \(B\) changes as a result of the collision, explaining your answer. Given that \(k = \frac { 7 } { 3 }\),
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse that \(A\) exerts on \(B\) in the collision.

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 \(\theta\)