Practice Paper Mechanics 2 (Edexcel International A Level Maths: Pure 3)

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 ladder , of length  m and mass 10 kg, is leaning against a smooth vertical wall at the point and resting on rough horizontal ground at the point ,   m from the wall.  The ladder lies in a vertical plane perpendicular to the wall and can be modelled as a uniform rod.  The coefficient of friction between the ladder and the ground is 0.45. 

When a box of tools of mass kg is placed four fifths of the way up the ladder, the ladder is in limiting equilibrium.  Find the value of  .

The box of tools is now moved further down the ladder to a point where the ladder is more stable and the magnitude of the reaction force at the point is exactly half of the reaction force at .  Given that the box of tools is now 2.8 m from the point , find the value of .

A company produces solid, flat brackets that give strength to pieces of furniture by cutting metal sheets into L-shapes and one such bracket is illustrated in the diagram below. 

(i) Find the exact coordinates of the centre of mass of a bracket.
 

(ii) Write down an assumption that is required for your answer to part (a) (i) to be valid.

In a quality control test a bracket is suspended from vertex  under its own weight.

Under the same assumption as used in part (a), determine the angle (to one decimal place) that a bracket makes between the downward vertical and the side .

In another quality control test the bracket has masses of 10 kg, 15 kg and 20 kg placed at the points respectively. The mass of the bracket is 1 kg.

Again, under the same assumption used in part (a), and assuming the bracket does not deform when the masses are added, find the exact coordinates of the centre of mass of the bracket when loaded with the three masses.

At time t seconds, a particle P has acceleration a m s⁻², where

                                                    .

Initially P starts at the origin O and moves with velocity .

Find the distance between the origin and the position P of when .

Find the value of t at the instant when P is moving in the direction of .

A car is moving up a straight hill, along the line of steepest slope, inclined at to the horizontal where .  The non-gravitational resistance force acting on the car parallel to the motion of the car has a magnitude of 0.115 , where  is the magnitude of the normal reaction force exerted on the car by the ground.  The car travels at a constant speed for 1 km and the work done by the engine of the car during this time is 5250 kJ. 

Find the combined mass of the car and its driver.

The car is travelling with speed 12 m s^-1 when the driver of the car sees a small dog run into the road just ahead and applies the brakes.  The total work done against the resistance force and the braking force is 21400 J.  The braking force acts parallel to the motion of the car and it can not be assumed to be constant.

Find the distance the car travels before coming to rest.

Two particles have masses and respectively, where is a constant.  are moving in the same direction along a straight line on a smooth horizontal plane when they collide directly.  Immediately before the collision,   has speed .  Immediately after the collision, the direction of motion of is reversed and it moves with speed  .  The speed of is .  The kinetic energy lost in the collision is .

Show that .

Show that the coefficient of restitution between is .

At the instant collide, they are a distance from a fixed vertical wall perpendicular to the direction of the motion of the particles.  After colliding with continues with constant speed until it hits the wall and rebounds.  A second collision between occurs a distance from the wall.

Find the coefficient of restitution between  and the wall.

A football is kicked from the top of a hill and its motion modelled as moving in a 2D plane under gravity. Its initial velocity is and it first hits ground level with velocity .

Find the height of the hill at the point from which the football was kicked.

Find the distance between the point from which the football was kicked and the point at which it first hits the ground.