Past Exam Questions

A bead, A, of mass 0.1 kg is threaded on a long straight rigid wire which is inclined at \(\sin^{-1}\left(\frac{7}{25}\right)\) to the horizontal. A is released from rest and moves down the wire. The coefficient of friction between A and the wire is \(\mu\). When A has travelled 0.45 m down the wire, its speed is 0.6 m s\(^{-1}\).

(a) Show that \(\mu = 0.25\).

Another bead, B, of mass 0.5 kg is also threaded on the wire. At the point where A has travelled 0.45 m down the wire, it hits B which is instantaneously at rest on the wire. A is brought to instantaneous rest in the collision. The coefficient of friction between B and the wire is 0.275.

(b) Find the time from when the collision occurs until A collides with B again.

A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120 kg and the mass of the post is 40 kg (see diagram). The object hits the post with speed 8 m/s and remains in contact with it after the impact.

(a) Calculate the speed with which the combined post and object moves immediately after the impact.

(b) There is a constant force resisting the motion of magnitude 4800 N. Calculate the distance the post is driven into the ground.

Particles P and Q have masses m kg and 2m kg respectively. The particles are initially held at rest 6.4 m apart on the same line of greatest slope of a rough plane inclined at an angle α to the horizontal, where \\sin α = 0.8\\ (see diagram). Particle P is released from rest and slides down the line of greatest slope. Simultaneously, particle Q is projected up the same line of greatest slope at a speed of 10 m s^-1. The coefficient of friction between each particle and the plane is 0.6.

1. Show that the acceleration of Q up the plane is \\-11.6\\ m s^-2.
2. Find the time for which the particles are in motion before they collide.
3. The particles coalesce on impact. Find the speed of the combined particle immediately after the impact.

Two small smooth spheres A and B, of equal radii and of masses \(km\) kg and \(m\) kg respectively, where \(k > 1\), are free to move on a smooth horizontal plane. A is moving towards B with speed 6 m s^-1 and B is moving towards A with speed 2 m s^-1. After the collision A and B coalesce and move with speed 4 m s^-1.

1. Find \(k\).
2. Find, in terms of \(m\), the loss of kinetic energy due to the collision.

Particles P of mass 0.4 kg and Q of mass 0.5 kg are free to move on a smooth horizontal plane. P and Q are moving directly towards each other with speeds 2.5 m s^-1 and 1.5 m s^-1 respectively. After P and Q collide, the speed of Q is twice the speed of P.

Find the two possible values of the speed of P after the collision.

A particle A is projected vertically upwards from level ground with an initial speed of 30 m s^-1. At the same instant a particle B is released from rest 15 m vertically above A. The mass of one of the particles is twice the mass of the other particle. During the subsequent motion A and B collide and coalesce to form particle C.

Find the difference between the two possible times at which C hits the ground.

Three particles P, Q and R, of masses 0.1 kg, 0.2 kg and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle P is projected towards Q at a speed of 5 m s^-1. After P and Q collide, P rebounds with speed 1 m s^-1.

1. Find the speed of Q immediately after the collision with P.
2. Q now collides with R. Immediately after the collision with Q, R begins to move with speed V m s^-1.
3. Given that there is no subsequent collision between P and Q, find the greatest possible value of V.

Two particles P and Q, of masses 6 kg and 2 kg respectively, lie at rest 12.5 m apart on a rough horizontal plane. The coefficient of friction between each particle and the plane is 0.4. Particle P is projected towards Q with speed 20 m/s^-1.

(a) Show that the speed of P immediately before the collision with Q is 10\(\sqrt{3}\) m/s^-1.

In the collision P and Q coalesce to form particle R.

(b) Find the loss of kinetic energy due to the collision.

The coefficient of friction between R and the plane is 0.4.

(c) Find the distance travelled by particle R before coming to rest.

Two particles P and Q of masses 0.2 kg and 0.3 kg respectively are free to move in a horizontal straight line on a smooth horizontal plane. P is projected towards Q with speed 0.5 m s^-1. At the same instant Q is projected towards P with speed 1 m s^-1. Q comes to rest in the resulting collision.

Find the speed of P after the collision.

Two small smooth spheres A and B, of equal radii and of masses 4 kg and m kg respectively, lie on a smooth horizontal plane. Initially, sphere B is at rest and A is moving towards B with speed 6 m s^-1. After the collision A moves with speed 1.5 m s^-1 and B moves with speed 3 m s^-1.

Find the two possible values of the loss of kinetic energy due to the collision.

Two particles P and Q, of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. P is projected towards Q with speed 2 m s^-1.

(a) Write down the momentum of P.

(b) After the collision P continues to move in the same direction with speed 0.3 m s^-1. Find the speed of Q after the collision.

A particle B of mass 5 kg is at rest on a smooth horizontal table. A particle A of mass 2.5 kg moves on the table with a speed of 6 m s^-1 and collides directly with B. In the collision, the two particles coalesce.

(a) Find the speed of the combined particle after the collision.

(b) Find the loss of kinetic energy of the system due to the collision.

Particles P of mass m kg and Q of mass 0.2 kg are free to move on a smooth horizontal plane. P is projected at a speed of 2 m s^-1 towards Q which is stationary. After the collision P and Q move in opposite directions with speeds of 0.5 m s^-1 and 1 m s^-1 respectively.

Find m.

Small smooth spheres A and B, of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially B is at rest and A is moving towards B with speed 10 m/s. After the spheres collide A continues to move in the same direction but with half the speed of B.

(a) Find the speed of B after the collision. [2]

A third small smooth sphere C, of mass 1 kg and with the same radius as A and B, is at rest on the plane. B now collides directly with C. After this collision B continues to move in the same direction but with one third the speed of C.

(b) Show that there is another collision between A and B. [3]

(c) A and B coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions. [5]

A particle P of mass 0.3 kg, lying on a smooth plane inclined at 30° to the horizontal, is released from rest. P slides down the plane for a distance of 2.5 m and then reaches a horizontal plane. There is no change in speed when P reaches the horizontal plane. A particle Q of mass 0.2 kg lies at rest on the horizontal plane 1.5 m from the end of the inclined plane (see diagram). P collides directly with Q.

(a) It is given that the horizontal plane is smooth and that, after the collision, P continues moving in the same direction, with speed 2 m s^-1.

Find the speed of Q after the collision.

(b) It is given instead that the horizontal plane is rough and that when P and Q collide, they coalesce and move with speed 1.2 m s^-1.

Find the coefficient of friction between P and the horizontal plane.

On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of 12 m s^-2. The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.

(a) Find the magnitude of the force in the tow-bar. [2]

(b) Find the braking force. [2]

(c) At the instant when the brakes are applied, the car has speed 22 m s^-1. At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of 8 m s^-1. [2]

(d) After the collision, the van starts to move with speed 5 m s^-1 and the car and trailer continue moving in the same direction with speed 2 m s^-1. Find the mass of the van. [3]

Two particles P and Q, of masses 0.1 kg and 0.4 kg respectively, are free to move on a smooth horizontal plane. Particle P is projected with speed 4 m/s^-1 towards Q which is stationary. After P and Q collide, the speeds of P and Q are equal.

Find the two possible values of the speed of P after the collision.

Two particles A and B, of masses 3.2 kg and 2.4 kg respectively, lie on a smooth horizontal table. A moves towards B with a speed of v m/s and collides with B, which is moving towards A with a speed of 6 m/s. In the collision the two particles come to rest.

(a) Find the value of v.

(b) Find the loss of kinetic energy of the system due to the collision.

Two particles P and Q, of masses m kg and 0.3 kg respectively, are at rest on a smooth horizontal plane. P is projected at a speed of 5 m s^-1 directly towards Q. After P and Q collide, P moves with a speed of 2 m s^-1 in the same direction as it was originally moving.

(a) Find, in terms of m, the speed of Q after the collision.

After this collision, Q moves directly towards a third particle R, of mass 0.6 kg, which is at rest on the plane. Q is brought to rest in the collision with R, and R begins to move with a speed of 1.5 m s^-1.

(b) Find the value of m.

The diagram shows a smooth track which lies in a vertical plane. The section AB is a quarter circle of radius 1.8 m with centre O. The section BC is a horizontal straight line of length 7.0 m and OB is perpendicular to BC. The section CFE is a straight line inclined at an angle of \(\theta\) above the horizontal.

A particle P of mass 0.5 kg is released from rest at A. Particle P collides with a particle Q of mass 0.1 kg which is at rest at B. Immediately after the collision, the speed of P is 4 m/s in the direction BC. You should assume that P is moving horizontally when it collides with Q.

(a) Show that the speed of Q immediately after the collision is 10 m/s.

When Q reaches C, it collides with a particle R of mass 0.4 kg which is at rest at C. The two particles coalesce. The combined particle comes instantaneously to rest at F. You should assume that there is no instantaneous change in speed as the combined particle leaves C, nor when it passes through C again as it returns down the slope.

(b) Given that the distance CF is 0.4 m, find the value of \(\theta\).

(c) Find the distance from B at which P collides with the combined particle.