6.03b Conservation of momentum: 1D two particles

Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m\) kg and \(km\) kg respectively, where \(k > 1\), move towards each other along a straight line and collide directly. Immediately before the collision, \(A\) has speed 5 ms\(^{-1}\) and \(B\) has speed 3 ms\(^{-1}\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude \(7km\) Ns.

1. Find the speed of \(B\) after the impact. [3 marks]
2. Show that the speed of \(A\) immediately after the collision is \((7k - 5)\) ms\(^{-1}\) and deduce that the direction of \(A\)'s motion is reversed. [5 marks]

\(B\) is now given a further impulse of magnitude \(mu\) Ns, as a result of which a second collision between it and \(A\) occurs.

1. Show that \(u > k(7k - 1)\). [4 marks]

Two trucks \(P\) and \(Q\), of masses 18 000 kg and 16 000 kg respectively, collide while moving towards each other in a straight line. Immediately before the collision, both trucks are travelling at the same speed, \(u\) ms\(^{-1}\). Immediately after the collision, \(P\) is moving at half its original speed, its direction of motion having been reversed.

1. Find, in terms of \(u\), the speed of \(Q\) immediately after the collision. \hfill [5 marks]
2. State, with a reason, whether the direction of \(Q\)'s motion has been reversed. \hfill [1 mark]
3. Find, in terms of \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision, stating the units of your answer. \hfill [3 marks]

The force exerted by each truck on the other in the impact has magnitude \(108000u\) N.

1. Find the time for which the trucks are in contact. \hfill [3 marks]

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 \(4\) m s\(^{-1}\) and \(Q\) has speed \(3\) m s\(^{-1}\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u\) m s\(^{-1}\) and \(Q\) with speed \((3.5 - u)\) m s\(^{-1}\).

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

After the collision the spheres both move with deceleration of magnitude \(5\) m s\(^{-2}\) until they come to rest on the plane.

1. Find the distance \(PQ\) when both \(P\) and \(Q\) are at rest. [4]

\includegraphics{figure_7} \(PQ\) is a line of greatest slope, of length \(4\) m, on a smooth plane inclined at \(30°\) to the horizontal. Particles \(A\) and \(B\), of masses \(0.15\) kg and \(0.5\) kg respectively, move along \(PQ\) with \(A\) below \(B\). The particles are both moving upwards, \(A\) with speed \(8\) m s\(^{-1}\) and \(B\) with speed \(2\) m s\(^{-1}\), when they collide at the mid-point of \(PQ\) (see diagram). Particle \(A\) is instantaneously at rest immediately after the collision.

1. Show that \(B\) does not reach \(Q\) in the subsequent motion. [8]
2. Find the time interval between the instant of \(A\)'s arrival at \(P\) and the instant of \(B\)'s arrival at \(P\). [6]

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

\includegraphics{figure_4} Three uniform spheres \(L\), \(M\) and \(N\) have masses \(0.8\) kg, \(0.6\) kg and \(0.7\) kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4\) m s\(^{-1}\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2\) m s\(^{-1}\) and \(0.5\) m s\(^{-1}\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2\) m s\(^{-1}\). Find the speed of \(L\) after the collision. [4]
2. \(M\) then collides with \(N\).
   1. Find the total momentum of \(M\) and \(N\) in the direction of \(M\)'s motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision. [4]
   2. Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision. [2]

\includegraphics{figure_4} Three uniform spheres \(L\), \(M\) and \(N\) have masses 0.8 kg, 0.6 kg and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4 \text{ m s}^{-1}\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2 \text{ m s}^{-1}\) and \(0.5 \text{ m s}^{-1}\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2 \text{ m s}^{-1}\). Find the speed of \(L\) after the collision. [4]
2. \(M\) then collides with \(N\).
   1. Find the total momentum of \(M\) and \(N\) in the direction of \(M\)'s motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision. [4]
   2. Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision. [2]

\includegraphics{figure_4} Four particles \(A\), \(B\), \(C\) and \(D\) are on the same straight line on a smooth horizontal table. \(A\) has speed \(6\text{ m s}^{-1}\) and is at rest towards \(B\). The speed of \(B\) is \(2\text{ m s}^{-1}\) and \(B\) is moving towards \(A\). The particle \(C\) is moving with speed \(5\text{ m s}^{-1}\) away from \(B\) and towards \(D\), which is stationary (see diagram). The first collision is between \(A\) and \(B\) which have masses \(0.8\text{ kg}\) and \(0.2\text{ kg}\) respectively.

1. After the particles collide \(A\) has speed \(4\text{ m s}^{-1}\) in its original direction of motion. Calculate the speed of \(B\) after the collision. [4]

The second collision is between \(C\) and \(D\) which have masses \(0.3\text{ kg}\) and \(0.1\text{ kg}\) respectively.

1. The particles coalesce when they collide. Find the speed of the combined particle after this collision. [3]

The third collision is between \(B\) and the combined particle, after which no further collisions occur.

1. Calculate the greatest possible speed of the combined particle after the third collision. [4]

\includegraphics{figure_5} Three uniform spheres \(A\), \(B\) and \(C\) have masses 0.3 kg, 0.4 kg and \(m\) kg respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \text{ m s}^{-1}\) towards \(B\) (see diagram). Air resistance may be ignored.

1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \text{ m s}^{-1}\). Find the speed with which \(B\) starts to move. [4]
2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \text{ m s}^{-1}\) and \(0.4 \text{ m s}^{-1}\) respectively. Find the value of \(m\). [4]
3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\). [1]
4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers. [4]

Two particles, \(P\) and \(Q\), of mass 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle \(P\) is projected away from \(Q\) with a speed of 7 ms\(^{-1}\).

1. Find the common speed of the particles after the string becomes taut. [3 marks]
2. Calculate the impulse in the string when it jerks tight. [2 marks]

A particle \(A\) of mass \(3m\) is moving along a straight line with constant speed \(u\) m s\(^{-1}\). It collides with a particle \(B\) of mass \(2m\) moving at the same speed but in the opposite direction. As a result of the collision, \(A\) is brought to rest.

1. Show that, after the collision, \(B\) has changed its direction of motion and that its speed has been halved. [4 marks]

Given that the magnitude of the impulse exerted by \(A\) on \(B\) is \(9m\) Ns,

1. find the value of \(u\). [3 marks]

A particle, \(P\), of mass 5 kg moves with speed 3 m s\(^{-1}\) along a smooth horizontal track. It strikes a particle \(Q\) of mass 2 kg which is at rest on the track. Immediately after the collision, \(P\) and \(Q\) move in the same direction with speeds \(v\) and 2v m s\(^{-1}\) respectively.

1. Calculate the value of \(v\). [3 marks]
2. Calculate the magnitude of the impulse received by \(Q\) on impact. [2 marks]

In a physics experiment, two balls \(A\) and \(B\), of mass \(4m\) and \(3m\) respectively, are travelling towards one another on a straight horizontal track. Both balls are travelling with speed 2 m s\(^{-1}\) immediately before they collide. As a result of the impact, \(A\) is brought to rest and the direction of motion of \(B\) is reversed. Modelling the track as smooth and the balls as particles,

1. find the speed of \(B\) immediately after the collision. [3 marks]

A student notices that after the collision, \(B\) comes to rest 0.2 m from \(A\).

1. Show that the coefficient of friction between \(B\) and the track is 0.113, correct to 3 decimal places. [7 marks]

Two particles \(P\) and \(Q\), of mass \(m\) and \(km\) respectively, are travelling in opposite directions on a straight horizontal path with speeds \(3u\) and \(2u\) respectively. \(P\) and \(Q\) collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.

1. Find the value of \(k\). [4 marks]
2. Write down an expression in terms of \(m\) and \(u\) for the magnitude of the impulse which \(P\) exerts on \(Q\) during the collision. [3 marks]

Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed 4 ms\(^{-1}\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed 2 ms\(^{-1}\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). Find

1. the speed of \(Q\) before the collision, [4 marks]
2. the speed of \(P\) after the collision, [4 marks]
3. the kinetic energy, in J, lost in the impact. [4 marks]

Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4u\) and \(5u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. [6 marks]
2. Show that \(e > \frac{1}{9}\). [2 marks]

\(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).

1. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur. [3 marks]

A small ball \(A\) is moving with velocity \((7\mathbf{i} + 12\mathbf{j})\) ms\(^{-1}\). It collides in mid-air with another ball \(B\), of mass \(0.4\) kg, moving with velocity \((-\mathbf{i} + 7\mathbf{j})\) ms\(^{-1}\). Immediately after the collision, \(A\) has velocity \((-3\mathbf{i} + 4\mathbf{j})\) ms\(^{-1}\) and \(B\) has velocity \((6.5\mathbf{i} + 13\mathbf{j})\) ms\(^{-1}\). Calculate the mass of \(A\). [4 marks]

Three particles \(A\), \(B\) and \(C\), of equal size and each of mass \(m\), are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(e\). \(A\) is projected with speed \(7\) ms\(^{-1}\) and strikes \(B\) directly. \(B\) then collides with \(C\), which starts to move with speed \(4\) ms\(^{-1}\). Calculate the value of \(e\). [10 marks]

Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.

1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]

Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).

1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]

Two railway trucks \(A\) and \(B\), whose masses are \(6m\) and \(5m\) respectively, are moving in the same direction along a straight track with speeds \(5u\) and \(3u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(kv\) respectively, in the same direction as before. The coefficient of restitution between \(A\) and \(B\) is \(e\). Modelling the trucks as particles,

1. show that
   1. \(v = \frac{45u}{5k + 6}\),
   2. \(v = \frac{2eu}{k - 1}\).
   [8 marks]
2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). [5 marks]

Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9m\) and \(4m\) respectively, are moving towards each other along a straight line with speeds 4 ms\(^{-1}\) and 6 ms\(^{-1}\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.

1. Show that the speed of \(B\) after the impact cannot be more than 3 ms\(^{-1}\). [5 marks]

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(e < \frac{3}{10}\). [5 marks]
2. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\). [4 marks]

\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).

1. Find the coefficient of restitution between \(A\) and \(B\). [4]
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]

Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.

1. Show that there will be another collision. [3]

A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find

1. the speed of \(A\) immediately after the collision, [4]
2. the distance \(A\) moves up the plane after the collision. [2]

The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.

1. Find the value of \(m\). [3]

A particle \(A\) of mass \(2m\) is moving with speed \(u\) on a smooth horizontal surface when it collides with a stationary particle \(B\) of mass \(m\). After the collision the speed of \(A\) is \(v\), the speed of \(B\) is \(3v\) and the particles move in the same direction.

1. Find \(v\) in terms of \(u\). [3]
2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\). [2]

\(B\) subsequently hits a vertical wall which is perpendicular to the direction of motion. As a result of the impact, \(B\) loses \(\frac{3}{4}\) of its kinetic energy.

1. Show that the speed of \(B\) after hitting the wall is \(\frac{3}{4}u\). [4]
2. \(B\) then hits \(A\). Calculate the speeds of \(A\) and \(B\), in terms of \(u\), after this collision and state their directions of motion. [8]