6.03b Conservation of momentum: 1D two particles

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. 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. 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.

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

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.

3. A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The blocks are modelled as particles.

1. Find the value of \(h\). Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
2. find the value of \(R\). \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-07_2252_51_315_36}
   

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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.

2 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-2_221_1153_1340_497} Three small uniform spheres \(A , B\) and \(C\) have masses \(0.4 \mathrm {~kg} , 1.2 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres move in the same straight line on a smooth horizontal table, with \(B\) between \(A\) and \(C\). Sphere \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 } , B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(C\) is moving towards \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Spheres \(A\) and \(B\) collide. After this collision \(B\) moves with speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(C\).

1. Find the speed with which \(A\) moves after the collision and state the direction of motion of \(A\).
2. Spheres \(B\) and \(C\) now collide and move away from each other with speeds \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).

2 An ice skater of mass 40 kg is moving in a straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when she collides with a skater of mass 60 kg moving in the opposite direction along the same straight line with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision the skaters move together with a common speed in the same straight line. Calculate their common speed, and state their direction of motion.

1 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_227_878_269_635} A particle \(P\) of mass 0.5 kg is travelling with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane towards a stationary particle \(Q\) of mass \(m \mathrm {~kg}\) (see diagram). The particles collide, and immediately after the collision \(P\) has speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that both particles are moving in the same direction after the collision, calculate \(m\).
2. Given instead that the particles are moving in opposite directions after the collision, calculate \(m\).

7 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles \(P\) and \(Q\) have masses 0.7 kg and 0.3 kg respectively. \(P\) and \(Q\) are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). Before \(P\) and \(Q\) collide the only horizontal force acting on each particle is friction and each particle decelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The particles coalesce when they collide.

1. Given that \(P\) and \(Q\) collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
2. Given instead that \(P\) and \(Q\) collide 3 s after projection,
   1. sketch on a single diagram the \(( t , v )\) graphs for the two particles in the interval \(0 \leqslant t < 3\),
   2. calculate the distance between the two particles at the instant when they are projected.

3 Two small spheres \(P\) and \(Q\) have masses 0.1 kg and 0.2 kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) with speed \(( 3.5 - u ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the value of \(u\). After the collision the spheres both move with deceleration of magnitude \(5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until they come to rest on the plane.
2. Find the distance \(P Q\) when both \(P\) and \(Q\) are at rest.

1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass \(m \mathrm {~kg}\) and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(m\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_149_606_1626_772} Two particles of masses 0.18 kg and m kg move on a smooth horizontal plane. They are moving towards each other in the same straight line when they collide. Immediately before the impact the speeds of the particles are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. Given that the particles are brought to rest by the impact, find m .
2. Given instead that the particles move with equal speeds of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after the impact, find
   1. the value of m , assuming that the particles move in opposite directions after the impact,
   2. the two possible values of m , assuming that the particles coalesce.

2 Two particles \(P\) and \(Q\) are moving in opposite directions in the same straight line on a smooth horizontal surface when they collide. \(P\) has mass 0.4 kg and speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 } . Q\) has mass 0.6 kg and speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the speed of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that \(P\) and \(Q\) are moving in the same direction after the collision, find the speed of \(Q\).
2. Given instead that \(P\) and \(Q\) are moving in opposite directions after the collision, find the distance between them 3 s after the collision.