6.03b Conservation of momentum: 1D two particles

2. Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. They are moving in opposite directions on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As a result of the collision, the direction of motion of \(B\) is reversed and its speed immediately after the collision is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) immediately after the collision, stating clearly whether the direction of motion of \(A\) is changed by the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision, stating clearly the units in which your answer is given.

2. Two particles \(A\) and \(B\), of mass 0.3 kg and \(m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line so that the particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. In the collision the direction of motion of each particle is reversed and, immediately after the collision, the speed of each particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

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

1. Particle \(P\) of mass \(m\) and particle \(Q\) of mass \(k m\) are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of \(P\) is \(5 u\) and the speed of \(Q\) is \(u\). Immediately after the collision the speed of each particle is halved and the direction of motion of each particle is reversed.

Find

1. the value of \(k\),
2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.

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.

6. A railway truck \(P\) of mass 1500 kg is moving on a straight horizontal track. The truck \(P\) collides with a truck \(Q\) of 2500 kg at a point \(A\). Immediately before the collision, \(P\) and \(Q\) are moving in the same direction with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after the collision, the direction of motion of \(P\) is unchanged and its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling the trucks as particles,

1. show that the speed of \(Q\) immediately after the collision is \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision at \(A\), the truck \(P\) is acted upon by a constant braking force of magnitude 500 N . The truck \(P\) comes to rest at the point \(B\).
2. Find the distance \(A B\). After the collision \(Q\) continues to move with constant speed \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance between \(P\) and \(Q\) at the instant when \(P\) comes to rest.

7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal surface with \(Q\) between \(P\) and \(R\). Particle \(P\) has mass \(m\), particle \(Q\) has mass \(2 m\) and particle \(R\) has mass \(3 m\). The coefficient of restitution between each pair of particles is \(e\). Particle \(P\) is projected towards \(Q\) with speed \(3 u\) and collides directly with \(Q\).

1. Find, in terms of \(u\) and \(e\),
   1. the speed of \(Q\) immediately after the collision,
   2. the speed of \(P\) immediately after the collision.
2. Find the range of values of \(e\) for which the direction of motion of \(P\) is reversed as a result of the collision with \(Q\). Immediately after the collision between \(P\) and \(Q\), particle \(R\) is projected towards \(Q\) with speed \(u\) so that \(R\) and \(Q\) collide directly. Given that \(e = \frac { 2 } { 3 }\)
3. show that there will be a second collision between \(P\) and \(Q\).

5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(3 m\) respectively, are moving in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly and, as a result of the collision, the direction of motion of \(P\) is reversed and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(\frac { 3 v } { 2 }\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\).

1. Find
   1. the speed of \(P\) immediately before the collision,
   2. the speed of \(Q\) immediately before the collision. After the collision with \(P\), the particle \(Q\) moves on the plane and strikes at right angles a fixed smooth vertical wall and rebounds. The coefficient of restitution between \(Q\) and the wall is \(e\). Given that there is a further collision between the particles,
2. find the range of possible values of \(e\).

7. A particle \(P\) of mass \(3 m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. A second particle \(Q\) of mass \(2 m\) is moving with speed \(2 u\) in the opposite direction to \(P\) along the same straight line. Particle \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the direction of motion of \(P\) is reversed as a result of the collision with \(Q\).
2. Find the range of values of \(e\) for which the direction of motion of \(Q\) is also reversed as a result of the collision. Given that \(e = \frac { 1 } { 2 }\)
3. find, in terms of \(m\) and \(u\), the kinetic energy lost in the collision between \(P\) and \(Q\).

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.

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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4. Two small balls, \(A\) and \(B\), are moving in opposite directions along the same straight line on smooth horizontal ground. The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\). The balls collide directly. Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\) By modelling the balls as particles,

1. show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 5 } u ( 1 + 6 e )\).
   (6) After the collision with ball \(A\), ball \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 5 } { 7 }\) Ball \(B\) rebounds from the wall and there is a second direct collision between \(A\) and \(B\).
2. Find the range of possible values of \(e\).

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 \(k < 5\)

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.

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.