Practice Paper 4 (Mechanics) (CIE A Level Maths: Pure 3)

Two spheres  and  are of equal radius and have masses  and  respectively.  and  are moving in the same direction in a straight line on a smooth horizontal surface when they collide directly. Immediately before the collision,  and  are moving with speeds  and  respectively. Immediately after the collision, the speed of  is , the speed of  is  and the direction of motion of  is unchanged.

Find the two possible values of .

Show that the minimum total kinetic energy lost in the collision is 

The following force diagram shows three forces acting on a particle:

Given that the resultant force on the particle in the vertical direction is 44.8 N downwards, find the size of the angle .  Give your answer in degrees correct to 3 significant figures.

A particle of mass 12 kg is being pushed up a smooth slope by a force of 50 N that acts horizontally. The slope is inclined at 20° to the horizontal, as shown in the diagram below:

Calculate the acceleration of the particle up the slope.

Calculate the normal reaction force of the slope on the particle.

A train consisting of an engine and five carriages moves forwards on a straight horizontal track.  The couplings that connect the carriages can be modelled as light and inextensible. A constant resistive force of 2800 N acts on the engine and a constant resistive force of 500 N acts on each of the five carriages.  The maximum speed of the train on the track is 35 m s^-1. 

Find the maximum power output of the engine.

Find the maximum acceleration of the train at the instant when it is travelling at a speed of 20 m s^-1.  The mass of the engine is 15 tonnes, and the mass of each carriage is 7 tonnes.

A home-made rocket is launched from rest at ground level with time .

The acceleration of the rocket, measured in metres per square second, is modelled by the equation

A particle moving along a straight line has velocity , at time t seconds, and its motion is described the equation

v = t² − 4t + 4           

Show that the acceleration of the particle is negative for the first 2 seconds of its motion.

Two particles A and B, of masses 2.7 kg and 2.2 kg respectively, are connected by means of a light inextensible string.  Particle A is held motionless on a rough fixed plane inclined at 25° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that B is hanging vertically downwards as shown in the diagram below:

The string between A and the pulley lies along a line of greatest slope of the plane, and B hangs freely from the pulley.  The coefficient of friction between particle A and the plane is μ.

The system is released from rest with the string taut.  Given that particle B descends 1.82 m in the first 3 seconds after it is released, find the value of μ.