Edexcel GCE Mechanics M2 June 2011 exam answers review

A car of mass 1000 kg moves with constant speed V m s⁻¹ up a straight road inclined at
an angle θ to the horizontal, where sin θ = 1/30. The engine of the car is working at a rate of 12 kW. The resistance to motion from non-gravitational forces has magnitude 500 N. Find the value of V.

(5)

A particle P of mass m is moving in a straight line on a smooth horizontal surface with speed 4u. The particle P collides directly with a particle Q of mass 3m which is at rest on the surface. The coefficient of restitution between P and Q is e. The direction of motion of P is reversed by the collision.

Show that e > 1/3

(8)

A ball of mass 0.5 kg is moving with velocity 12i m s^–1 when it is struck by a bat. The impulse received by the ball is (–4i + 7j) Ns. By modelling the ball as a particle, find

1. the speed of the ball immediately after the impact,(4)
2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector i,(2)
3. the kinetic energy gained by the ball as a result of the impact.(2)

Figure 1 shows a uniform lamina ABCDE such that ABDE is a rectangle, BC = CD, AB = 4a and AE = 2a. The point F is the midpoint of BD and FC = a.

1. Find, in terms of a, the distance of the centre of mass of the lamina from AE.(4)

The lamina is freely suspended from A and hangs in equilibrium.

2. Find the angle between AB and the downward vertical.(3)

A particle P of mass 0.5 kg is projected from a point A up a line of greatest slope AB of a fixed plane. The plane is inclined at 30° to the horizontal and AB = 2 m with B above A, as shown in Figure 2. The particle P passes through B with speed 5 m s^–1. The plane is smooth from A to B.

1. Find the speed of projection.(4)

The particle P comes to instantaneous rest at the point C on the plane, where C is above B and BC = 1.5 m. From B to C the plane is rough and the coefficient of friction between P and the plane is µ.

By using the work-energy principle,

2. find the value of µ.(6)

A particle P moves on the x-axis. The acceleration of P at time t seconds is (t − 4) m s⁻² in the positive x-direction. The velocity of P at time t seconds is v m s^–1. When t = 0, v = 6.

Find

1. v in terms of t, (4)
2. the values of t when P is instantaneously at rest, (3)
3. the distance between the two points at which P is instantaneously at rest.(4)

A uniform rod AB, of mass 3m and length 4a, is held in a horizontal position with the end A against a rough vertical wall. One end of a light inextensible string BD is attached to the rod at B and the other end of the string is attached to the wall at the point D vertically above A, where AD = 3a. A particle of mass 3m is attached to the rod at C, where AC = x. The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is 25/4mg

Show that

1. x = 3a,(5)
2. the horizontal component of the force exerted by the wall on the rod has magnitude 5mg.(3)

The coefficient of friction between the wall and the rod is µ. Given that the rod is about to slip,

3. find the value of µ.(5)

A particle is projected from a point O with speed u at an angle of elevation α above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance x, its height above O is y.

1. Show that

   y = xtan α – gx²/2u²cos²α

   (4)

A girl throws a ball from a point A at the top of a cliff. The point A is 8 m above a horizontal beach. The ball is projected with speed 7 m s⁻¹ at an angle of elevation of 45°. By modelling the ball as a particle moving freely under gravity,

2. find the horizontal distance of the ball from A when the ball is 1 m above the beach. 5

A boy is standing on the beach at the point B vertically below A. He starts to run in a straight line with speed v m s−1, leaving B 0.4 seconds after the ball is thrown.

He catches the ball when it is 1 m above the beach.

3. Find the value of v.(4)