Practice Paper Mechanics 2 (Edexcel International A Level Maths: Pure 4)

The position vector of a particle at time  seconds is given by

Find an expression for the velocity of the particle at time seconds.

A uniform lamina is made using a uniform circular disc, made from a uniform material, with a rectangular hole cut from it. Representing the disc on the plane the circle has equation and the rectangle has vertices at the intersections of the four lines , as shown in the diagram below.

Find the exact coordinates of the centre of mass of the disc after the rectangular hole has been removed from it.

The lamina is suspended freely from the point . Show that the angle the downward vertical makes with the radius of the circle to the point is less than 0.5°.

A particle of mass 6 kg is moving with speed 12 m s^-1 in a horizontal plane when it receives an impulse.  Immediately after receiving the impulse,  is moving with speed 25 m s^-1 at an angle of 45° to the horizontal as shown in the diagram above.

Find the magnitude of the impulse.

Find the acute angle between the direction of the impulse and the horizontal.

A non-uniform rod of length 2 m and weight 20 N is resting on two parallel pegs at points and such that 0.8 m and 1.8 m.  The rod is resting on rough ground at an angle of with the horizontal such that  and is in limiting equilibrium.  The coefficient of friction between the rod and each peg is 0.3 and between the rod and the ground is 0.4.  

Given that the normal reaction at the point is twice the normal reaction at the point , calculate the normal reaction force at each peg and the ground.

Find the distance of the centre of mass of the rod from the point .

A toboggan team consisting of two children and their sled are moving down a 400 m long uneven course of varying gradients.  They start from rest at the top of the course and do no work as they move down the course.  The frictional force on the sled is 8 N and all other resistance forces can be ignored.  The vertical height of the hill is 35 m.  Find the speed that the team is moving at when they reach the end of the course.  You may consider the two children and their sled as a single particle of mass 100kg.

Two children in a sledge are being pulled up a smooth hill inclined at 10° to the horizontal by a light, inextensible rope inclined at 30° to the direction of motion.  The tension in the rope as it is being pulled has magnitude 120 N and the sledge moves forward 15 metres along the line of greatest slope from the point to the point .

Draw a diagram showing the motion of the sledge between the points and , labelling all forces acting on the sledge.

Calculate the total work done by the tension in the rope as the sledge is moved up the slope from to .

Once the sledge reaches point B it is then released from rest and accelerates down the hill.  At the point where the sledge has 440 J of kinetic energy, it is moving with speed of 4 m s^-1.  Find the total mass of the sledge and the two children.

Find the speed of the sledge at the instant when it has reached the point where it started at the bottom of the hill.

State two assumptions you have made in your answer to part (c)

A golfer strikes a ball from ground level with velocity  at an angle of  to the horizontal.

Write down the initial velocity of the golf ball in the form , giving the values of  as exact values.

A tree of height  stands in the path of the flight of the golf ball  from the point where it is struck.  Determine whether or not the golf ball strikes the tree.

For a particle modelled as a projectile with initial velocity  m s⁻¹ at an angle of  above the horizontal, show that the equation of the trajectory of the particle is given by

Two particles with masses 0.5 kg and 0.9 kg respectively are moving towards each other in opposite directions along a straight line on a smooth horizontal surface.  They collide directly. Immediately before the collision,  has speed 5 m s^-1 and  has speed 7 m s^-1.  The coefficient of restitution between is .

Find the speed of and the speed of  immediately after the collision.  Clearly state the direction of motion of each particle.

Calculate the kinetic energy lost in the collision.

Give one form of energy into which the lost kinetic energy could have been transformed.