Edexcel A Level Maths M2 · Mechanics 2 June 2009 question paper 6678

A particle of mass 0.25 kg is moving with velocity (3i + 7j) m s^–1 when it receives the impulse (5 i – 3 j) N s. Find the speed of the particle immediately after the impulse.

At time t = 0 a particle P leaves the origin O and moves along the x- axis. At time t seconds the velocity of P is v m s^–1, where

Find the maximum value of v.

Find the time taken for P to return to O.

A truck of mass of 300 kg moves along a straight horizontal road with a constant speed of 10 m s^–1. The resistance to motion of the truck has magnitude 120 N.

Find the rate at which the engine of the truck is working.

On another occasion the truck moves at a constant speed up a hill inclined at θ to the horizontal, where sin θ . The resistance to motion of the truck from non-gravitational forces remains of magnitude 120 N. The rate at which the engine works is the same as in part (a).

Find the speed of the truck.

A uniform rod AB, of length 1.5 m and mass 3 kg, is smoothly hinged to a vertical wall at A. The rod is held in equilibrium in a horizontal position by a light strut CD as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The end C of the strut is freely jointed to the wall at a point 0.5 m vertically below A. The end D is freely joined to the rod so that AD is 0.5 m.

Find the thrust in CD.

Find the magnitude and direction of the force exerted on the rod AB at A.

A shop sign ABCDEFG is modelled as a uniform lamina, as illustrated in Figure 2. ABCD is a rectangle with BC = 120 cm and DC = 90 cm. The shape EFG is an isosceles triangle with EG = 60 cm and height 60 cm. The mid-point of AD and the mid-point of EG coincide.

Find the distance of the centre of mass of the sign from the side AD.

The sign is freely suspended from A and hangs at rest.

Find the size of the angle between AB and the vertical.

A child playing cricket on horizontal ground hits the ball towards a fence 10 m away. The ball moves in a vertical plane which is perpendicular to the fence. The ball just passes over the top of the fence, which is 2 m above the ground, as shown in Figure 3. The ball is modelled as a particle projected with initial speed u m s^–1 from point O on the ground at an angle α to the ground.

By writing down expressions for the horizontal and vertical distances, from O of the ball t seconds after it was hit, show that

Given that α = 45°,

find the speed of the ball as it passes over the fence.

A particle P of mass 2 kg is projected up a rough plane with initial speed 14 m s^–1, from a point X on the plane, as shown in Figure 4. The particle moves up the plane along the line of greatest slope through X and comes to instantaneous rest at the point Y. The plane is inclined at an angle α to the horizontal, where tan α = . The coefficient of friction between the particle and the plane is .

Use the work-energy principle to show that XY = 25 m.

After reaching Y, the particle P slides back down the plane.

Find the speed of P as it passes through X.

Particles A, B and C of masses 4 m, 3 m and m respectively, lie at rest in a straight line on a smooth horizontal plane with B between A and C. Particles A and B are projected towards each other with speeds u m s^–1 and v m s^–1 respectively, and collide directly. As a result of the collision, A is brought to rest and B rebounds with speed k v m s^–1. The coefficient of restitution between A and B is

Show that u = 3 v.

Find the value of k.

Immediately after the collision between A and B, particle C is projected with speed 2v m s^–1 towards B so that B and C collide directly.

Show that there is no further collision between A and B.