6.03b Conservation of momentum: 1D two particles

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 \(km\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2u\). 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 \(k\). [4]
2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision. [2]

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 m s\(^{-1}\). The blocks are modelled as particles.

1. Find the value of \(h\). [2] 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 \(R\). [8]

Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) 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 \(2u\) and the speed of \(Q\) is \(4u\). In the collision, the particles join together to form a single particle. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision. [6]

A railway truck \(S\) of mass 20 tonnes is moving along a straight horizontal track when it collides with another railway truck \(T\) of mass 30 tonnes which is at rest. Immediately before the collision the speed of \(S\) is \(4\text{ ms}^{-1}\) As a result of the collision, the two railway trucks join together. Find

1. the common speed of the railway trucks immediately after the collision, [2]
2. the magnitude of the impulse exerted on \(S\) in the collision, stating the units of your answer. [3]

Particle \(P\) has mass \(m\) kg and particle \(Q\) has mass \(3m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4u\) m s\(^{-1}\) and \(Q\) has speed \(ku\) m s\(^{-1}\), where \(k\) is a constant. 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 \(k\). [4]
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\). [3]

A railway truck \(A\) of mass 1800 kg is moving along a straight horizontal track with speed 4 m s\(^{-1}\). It collides directly with a stationary truck \(B\) of mass 1200 kg on the same track. In the collision, \(A\) and \(B\) are coupled and move off together.

1. Find the speed of the trucks immediately after the collision. [3]

After the collision, the trucks experience a constant resistive force of magnitude \(R\) newtons. They come to rest 8 s after the collision.

1. Find \(R\). [3]

A railway truck \(P\) of mass 2000 kg is moving along a straight horizontal track with speed 10 m s\(^{-1}\). The truck \(P\) collides with a truck \(Q\) of mass 3000 kg, which is at rest on the same track. Immediately after the collision \(Q\) moves with speed 5 m s\(^{-1}\). Calculate

1. the speed of \(P\) immediately after the collision, [3]
2. the magnitude of the impulse exerted by \(P\) on \(Q\) during the collision. [2]

Two trucks \(A\) and \(B\), moving in opposite directions on the same horizontal railway track, collide. The mass of \(A\) is 600 kg. The mass of \(B\) is \(m\) kg. Immediately before the collision, the speed of \(A\) is 4 m s\(^{-1}\) and the speed of \(B\) is 2 m s\(^{-1}\). Immediately after the collision, the trucks are joined together and move with the same speed 0.5 m s\(^{-1}\). The direction of motion of \(A\) is unchanged by the collision. Find

1. the value of \(m\), [4]
2. the magnitude of the impulse exerted on \(A\) in the collision. [3]

A particle \(P\) of mass 1.5 kg is moving along a straight horizontal line with speed 3 m s\(^{-1}\). Another particle \(Q\) of mass 2.5 kg is moving, in the opposite direction, along the same straight line with speed 4 m s\(^{-1}\). The particles collide. Immediately after the collision the direction of motion of \(P\) is reversed and its speed is 2.5 m s\(^{-1}\).

1. Calculate the speed of \(Q\) immediately after the impact. [3]
2. State whether or not the direction of motion of \(Q\) is changed by the collision. [1]
3. Calculate the magnitude of the impulse exerted by \(Q\) on \(P\), giving the units of your answer. [3]

A particle \(P\) of mass 0.3 kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal table. The particle \(P\) collides directly with a particle \(Q\) of mass 0.6 kg, which is at rest on the table. Immediately after the particles collide, \(P\) has speed 2 m s\(^{-1}\) and \(Q\) has speed 5 m s\(^{-1}\). The direction of motion of \(P\) is reversed by the collision. Find

1. the value of \(u\), [4]
2. the magnitude of the impulse exerted by \(P\) on \(Q\). [2]

Immediately after the collision, a constant force of magnitude \(R\) newtons is applied to \(Q\) in the direction directly opposite to the direction of motion of \(Q\). As a result \(Q\) is brought to rest in 1.5 s.

1. Find the value of \(R\). [4]

A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed 12 m s\(^{-1}\). Another particle \(B\) of mass \(m\) kg is moving along the same straight line, in the opposite direction to \(A\), with speed 8 m s\(^{-1}\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed 3 m s\(^{-1}\) and \(B\) is moving with speed 4 m s\(^{-1}\). Find

1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision, [2]
2. the value of \(m\). [4]

Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2u\) and \(5u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac{1}{2}u\) and its direction of motion is reversed.

1. Find the speed and direction of motion of \(Q\) after the collision. [4]
2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision. [3]

The masses of two particles \(A\) and \(B\) are \(0.5 \text{ kg}\) and \(m \text{ kg}\) respectively. The particles are moving on a smooth horizontal table in opposite directions and collide directly. Immediately before the collision the speed of \(A\) is \(5 \text{ m s}^{-1}\) and the speed of \(B\) is \(3 \text{ m s}^{-1}\). In the collision, the magnitude of the impulse exerted by \(B\) on \(A\) is \(3.6 \text{ Ns}\). As a result of the collision the direction of motion of \(A\) is reversed.

1. Find the speed of \(A\) immediately after the collision. [3]

The speed of \(B\) immediately after the collision is \(1 \text{ m s}^{-1}\).

1. Find the two possible values of \(m\). [4]

A particle \(P\) of mass \(2\) kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4\) kg which is at rest on the same horizontal plane. Immediately after the collision, \(P\) and \(Q\) are moving in opposite directions and the speed of \(P\) is one-third the speed of \(Q\).

1. Show that the speed of \(P\) immediately after the collision is \(\frac{1}{5}u\) m s\(^{-1}\). [4]

After the collision \(P\) continues to move in the same straight line and is brought to rest by a constant resistive force of magnitude \(10\) N. The distance between the point of collision and the point where \(P\) comes to rest is \(1.6\) m.

1. Calculate the value of \(u\). [5]

Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \text{ m s}^{-1}\) and the speed of \(B\) is \(2 \text{ m s}^{-1}\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find

1. the speed of \(A\) immediately after the collision, [5]
2. the magnitude of the impulse exerted on \(B\) in the collision. [3]

Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(2m\) 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 \(2u\) and the speed of \(B\) is \(3u\). The magnitude of the impulse received by each particle in the collision is \(\frac{7mu}{2}\). Find

1. the speed of \(A\) immediately after the collision, [3]
2. the speed of \(B\) immediately after the collision. [3]

Particle \(P\) has mass 3 kg and particle \(Q\) has mass 2 kg. The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, \(P\) has speed 3 m s\(^{-1}\) and \(Q\) has speed 2 m s\(^{-1}\). Immediately after the collision, both particles move in the same direction and the difference in their speeds is 1 m s\(^{-1}\).

1. Find the speed of each particle after the collision. [5]
2. Find the magnitude of the impulse exerted on \(P\) by \(Q\). [3]

Two particles \(A\) and \(B\), of mass 2 kg and 3 kg respectively, 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 \(A\) is 5 m s\(^{-1}\) and the speed of \(B\) is 6 m s\(^{-1}\). The magnitude of the impulse exerted on \(B\) by \(A\) is 14 N s. Find

1. the speed of \(A\) immediately after the collision, [3]
2. the speed of \(B\) immediately after the collision. [3]

Particle \(P\) has mass 3 kg and particle \(Q\) has mass \(m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of \(P\) is \(4 \text{ m s}^{-1}\) and the speed of \(Q\) is \(3 \text{ m s}^{-1}\). In the collision the direction of motion of \(P\) is unchanged and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(1 \text{ m s}^{-1}\) and the speed of \(Q\) is \(1.5 \text{ m s}^{-1}\).

1. Find the magnitude of the impulse exerted on \(P\) in the collision. [3]
2. Find the value of \(m\). [3]

A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of \(1.6\) m above the top of the post.

1. Show that the speed of the pile-driver just before it hits the post is \(5.6\) m s\(^{-1}\). [2]

The post has mass \(6\) kg and the pile-driver has mass \(78\) kg. When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.

1. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. [3]

The post is brought to rest by the action of a resistive force from the ground acting for \(0.06\) s. By modelling this force as constant throughout this time,

1. find the magnitude of the resistive force, [4]
2. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest. [4]

A railway truck \(S\) of mass 2000 kg is travelling due east along a straight horizontal track with constant speed 12 m s\(^{-1}\). The truck \(S\) collides with a truck \(T\) which is travelling due west along the same track as \(S\) with constant speed 6 m s\(^{-1}\). The magnitude of the impulse of \(T\) on \(S\) is 28800 Ns.

1. Calculate the speed of \(S\) immediately after the collision. [3]
2. State the direction of motion of \(S\) immediately after the collision. [1]

Given that, immediately after the collision, the speed of \(T\) is 3.6 m s\(^{-1}\), and that \(T\) and \(S\) are moving in opposite directions,

1. calculate the mass of \(T\). [4]

A tent peg is driven into soft ground by a blow from a hammer. The tent peg has mass 0.2 kg and the hammer has mass 3 kg. The hammer strikes the peg vertically. Immediately before the impact, the speed of the hammer is \(16 \text{ m s}^{-1}\). It is assumed that, immediately after the impact, the hammer and the peg move together vertically downwards.

1. Find the common speed of the peg and the hammer immediately after the impact. [3]

Until the peg and hammer come to rest, the resistance exerted by the ground is assumed to be constant and of magnitude \(R\) newtons. The hammer and peg are brought to rest 0.05 s after the impact.

1. Find, to 3 significant figures, the value of \(R\). [5]

A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \text{ m s}^{-1}\), the speed of \(B\) is \(4 \text{ m s}^{-1}\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.

1. Find the speed and direction of \(C\) after the collision. [4]
2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision. [3]
3. State a modelling assumption which you have made about the trucks in your solution [1]

Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.

1. Find the value of \(d\). [4]

Three particles \(A\), \(B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5u\) and \(4u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3u\). In the subsequent motion, \(A\), \(B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. [7]

Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),

1. find the set of possible values of \(e\). [4]