Edexcel A Level Maths M2 · Mechanics 2 January 2011 question paper 6678

A cyclist starts from rest and moves along a straight horizontal road. The combined mass of the cyclist and his cycle is 120 kg. The resistance to motion is modelled as a constant force of magnitude 32 N. The rate at which the cyclist works is 384 W. The cyclist accelerates until he reaches a constant speed of v m s^–1. Find

the value of v,

the acceleration of the cyclist at the instant when the speed is 9 m s^–1.

A particle of mass 2 kg is moving with velocity (5i + j) m s^–1 when it receives an impulse of (–6 i + 8 j) N s. Find the kinetic energy of the particle immediately after receiving the impulse.

A particle moves along the x-axis. At time t = 0 the particle passes through the origin with speed 8 m s^–1 in the positive x-direction. The acceleration of the particle at time t seconds, t 0, is (4t³ – 12t) m s^–2 in the positive x-direction. Find

the velocity of the particle at time t seconds,

the displacement of the particle from the origin at time t seconds,

the values of t at which the particle is instantaneously at rest.

A box of mass 30 kg is held at rest at point A on a rough inclined plane. The plane is inclined at 20° to the horizontal. Point B is 50 m from A up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from A to B by a force acting parallel to AB and then held at rest at B. The coefficient of friction between the box and the plane is . Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,

find the work done in dragging the box from A to B.

The box is released from rest at the point B and slides down the slope. Using the work-energy principle, or otherwise,

find the speed of the box as it reaches A.

The uniform L-shaped lamina ABCDEF, shown in Figure 2, has sides AB and FE parallel, and sides BC and ED parallel. The pairs of parallel sides are 9 cm apart. The points A, F, D and C lie on a straight line. AB = BC = 36 cm, FE = ED = 18 cm. ABC = FED = 90°, and BCD = EDF = EFD = BAC = 45°.

Find the distance of the centre of mass of the lamina from

side AB,

side BC.

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

Find, to the nearest degree, the size of the angle between AB and the vertical.

[ In this question, the unit vectors i and j are in a vertical plane, i being horizontal and j being vertically upwards. ]

At time t = 0, a particle P is projected from the point A which has position vector 10 j metres with respect to a fixed origin O at ground level. The ground is horizontal. The velocity of projection of P is (3i + 5j) m s^–1, as shown in Figure 3. The particle moves freely under gravity and reaches the ground after T seconds.

For 0 tT, show that, with respect to O, the position vector, r metres, of P at time t seconds is given by

Find the value of T.

Find the velocity of P at time t seconds (0 t T).

When P is at the point B, the direction of motion of P is 45° below the horizontal.

Find the time taken for P to move from A to B.

Find the speed of P as it passes through B.

A uniform plank AB, of weight 100 N and length 4 m, rests in equilibrium with the end A on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is C, where AC = 3 m, as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angleto the horizontal, where sin = The coefficient of friction between the plank and the ground is . Modelling the plank as a rod, find the least possible value of .

A particle P of mass m kg is moving with speed 6 m s^–1 in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J. The coefficient of restitution between P and the wall is

Show that m = 4.

After rebounding from the wall, P collides directly with a particle Q which is moving towards P with speed 3 m s^–1. The mass of Q is 2 kg and the coefficient of restitution

Show that there will be a second collision between P and the wall.