Edexcel A Level Maths M2 · Mechanics 2 June 2008 question paper 6678

A lorry of mass 2000 kg is moving down a straight road inclined at angle to the horizontal, where sin=αsin = . The resistance to motion is modelled as a constant force of magnitude 1600 N. The lorry is moving at a constant speed of 14 m s^–1. Find, in kW, the rate at which the lorry’s engine is working.

A particle A of mass 4 m is moving with speed 3 u in a straight line on a smooth horizontal table. The particle A collides directly with a particle B of mass 3 m moving with speed 2 u in the same direction as A. The coefficient of restitution between A and B is e. Immediately after the collision the speed of B is 4 eu.

Find the total kinetic energy lost in the collision.

A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of 20° to the horizontal. The package slides down a line of greatest slope of the plane from a point A to a point B, where AB = 14 m. At A the package has speed 12 m s^–1 and at B the package has speed 8 m s^–1, as shown in Figure 1. Find

the total energy lost by the package in travelling from A to B,

the coefficient of friction between the package and the ramp.

A particle P of mass 0.5 kg is moving under the action of a single force F newtons. At time t seconds,

The velocity of P at time t seconds is v m s^–1. When t = 0, v = i – 4j.

Find v at time t seconds.

When t = 3, the particle P receives an impulse (–5 i + 12 j) N s.

Find the speed of P immediately after it receives the impulse.

Figure 2

A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod AB and the pole as a smooth horizontal peg perpendicular to the vertical plane containing AB. The rod has length 3 a and weight W and rests on the peg at C, where AC = 2 a. The end A of the rod rests on rough horizontal ground and AB makes an angle with the ground, as shown in Figure 2.

Show that the normal reaction on the rod at A is 4(4(4 .32cos α W)32cos cos² ) W Given that the rod is in limiting equilibrium and that cos α =cos =

find the coefficient of friction between the rod and the ground.

Figure 3

Figure 3 shows a rectangular lamina OABC. The coordinates of O, A, B and C are (0, 0), (8, 0), (8, 5) and (0, 5) respectively. Particles of mass km, 5 m and 3 m are attached to the lamina at A, B and C respectively. The x-coordinate of the centre of mass of the three particles without the lamina is 6.4.

Show that k = 7.

The lamina OABC is uniform and has mass 12 m.

Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina.

The combined system is freely suspended from O and hangs at rest.

Find the angle between OC and the horizontal.

A ball is thrown from a point A at a target, which is on horizontal ground. The point A is 12 m above the point O on the ground. The ball is thrown from A with speed 25 m s^–1 at an angle of 30° below the horizontal. The ball is modelled as a particle and the target as a point T. The distance OT is 15 m. The ball misses the target and hits the ground at the point B, where OTB is a straight line, as shown in Figure 4. Find

the time taken by the ball to travel from A to B,

the distance TB.

The point X is on the path of the ball vertically above T.

Find the speed of the ball at X.