Edexcel A Level Maths M2 · Mechanics 2 June 2016 question paper 6678

A particle P moves along a straight line. The speed of P at time t seconds (t 0) is v m s^–1, where v = (pt² + qt + r) and p, q and r are constants. When t = 2 the speed of P has its minimum value. When t = 0, v = 11 and when t = 2, v = 3 Find

the acceleration of P when t = 3

the distance travelled by P in the third second of the motion.

A car of mass 800kg is moving on a straight road which is inclined at an angle to the horizontal, where sin θ . The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude R newtons. When the car is moving up the road at a constant speed of 12.5ms, the engine of the car is working at a constant rate of 3 P watts. When the car is moving down the road at a constant speed of 12.5ms, the engine of the car is working at a constant rate of P watts.

Find

the value of P,

the value of R.

When the car is moving up the road at 12.5ms the engine is switched off and the car comes to rest, without braking, in a distance d metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude R newtons.

Use the work-energy principle to find the value of d.

A particle of mass 0.6kg is moving with constant velocity (ci + 2cj)ms, where c

Immediately after receiving the impulse the particle has velocity (2ci – cj)ms. Find the value of c.

The uniform lamina OBC is one quarter of a circular disc with centre O and radius 4m. The points A and D, on OB and OC respectively, are 3m from O. The uniform lamina ABCD, shown shaded in Figure 1, is formed by removing the triangle OAD from OBC. Given that the centre of mass of one quarter of a uniform circular disc of radius r is at a

find the distance of the centre of mass of the lamina ABCD from AD.

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

Find, to the nearest degree, the angle between DC and the downward vertical.

A non-uniform rod AB, of mass 5kg and length 4m, rests with one end A on rough horizontal ground. The centre of mass of the rod is d metres from A. The rod is held in limiting equilibrium at an angle to the horizontal by a force P, which acts in a direction perpendicular to the rod at B, as shown in Figure 2. The line of action of P lies in the same vertical plane as the rod.

Find, in terms of d, g and ,

the magnitude of the vertical component of the force exerted on the rod by the ground,

the magnitude of the friction force acting on the rod at A.

Given that tan θ and that the coefficient of friction between the rod and the ground

find the value of d.

[ In this question, i is a horizontal unit vector and j is an upward vertical unit vector.] A particle P is projected from a fixed origin O with velocity (3i + 4j)m s. The particle moves freely under gravity and passes through the point A with position vector (i – j)m, where is a positive constant.

Find

the speed of P at the instant when it passes through A,

the direction of motion of P at the instant when it passes through A.

Two particles A and B, of mass 2 m and 3 m respectively, are initially at rest on a smooth horizontal surface. Particle A is projected with speed 3u towards B. Particle A collides directly with particle B. The coefficient of restitution between A and B is

Find

the speed of A immediately after the collision,

the speed of B immediately after the collision.

After the collision B hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of B. The coefficient of restitution between B and the wall is e. The magnitude of the impulse received by B when it hits the wall is

Find the value of e.

Determine whether there is a further collision between A and B after B rebounds from the wall.