Edexcel A Level Maths M1 · Mechanics 1 June 2017 question paper 6677

Three forces, (15 i + j) N, (5 q i – p j) N and (–3 p i – q j) N, where p and q are constants, act on a particle. Given that the particle is in equilibrium, find the value of p and the value of q.

Two particles, P and Q, have masses 2 m and 3 m respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of P is 4 u and the speed of Q is 3 u. As a result of the collision, Q has its direction of motion reversed and is moving with speed u.

Find the speed of P immediately after the collision.

State whether or not the direction of motion of P has been reversed by the collision.

Find the magnitude of the impulse exerted on P by Q in the collision.

A plank AB has length 6m and mass 30kg. The point C is on the plank with CB = 2m. The plank rests in equilibrium in a horizontal position on supports at A and C. Two people, each of mass 75kg, stand on the plank. One person stands at the point P of the plank, where AP = x metres, and the other person stands at the point Q of the plank, where AQ = 2 x metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at C five times the magnitude of the reaction at A. The plank is modelled as a uniform rod and each person is modelled as a particle.

Find the value of x.

State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.

A particle P of mass 5kg is held at rest in equilibrium on a rough inclined plane by a horizontal force of magnitude 10N. The plane is inclined to the horizontal at an angle α where tan , as shown in Figure 1. The line of action of the force lies in the vertical plane containing P and a line of greatest slope of the plane. The coefficient of friction between P and the plane is Given that P is on the point of sliding down the plane, find the value of

A vertical light rod PQ has a particle of mass 0.5kg attached to it at P and a particle of mass 0.75kg attached to it at Q, to form a system, as shown in Figure 2. The system is accelerated vertically upwards by a vertical force of magnitude 15N applied to the particle at Q. Find the thrust in the rod.

A cyclist is moving along a straight horizontal road and passes a point A. Five seconds later, at the instant when she is moving with speed 10 ms^–1, she passes the point She moves with constant acceleration from A to B. Given that AB = 40m, find

the acceleration of the cyclist as she moves from A to B,

the time it takes her to travel from A to the midpoint of

[ In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.] Two ships, P and Q, are moving with constant velocities. The velocity of P is (9i – 2j)km h^–1 and the velocity of Q is (4i + 8j) km h^–1

Find the direction of motion of P, giving your answer as a bearing to the nearest degree.

When t = 0, the position vector of P is (9 i + 10 j)km and the position vector of Q is (i + 4 j) km. At time t hours, the position vectors of P and Q are p km and q km respectively.

Find an expression for

p in terms of t,

q in terms of t.

Hence show that, at time t hours,

Find the values of t when the ships are 10km apart.

Two particles, A and B, have masses 2 m and m respectively. The particles are attached to the ends of a light inextensible string. Particle A is held at rest on a fixed rough horizontal table at a distance d from a small smooth light pulley which is fixed at the edge of the table at the point P. The coefficient of friction between A and the table is , where < The string is parallel to the table from A to P and passes over the pulley. Particle B hangs freely at rest vertically below P with the string taut and at a height h, (h < d), above a horizontal floor, as shown in Figure 3. Particle A is released from rest with the string taut and slides along the table.

Write down an equation of motion for

Write down an equation of motion for

Hence show that, until B hits the floor, the acceleration of A is 1 2μ).

Find, in terms of g, h and μ, the speed of A at the instant when B hits the floor.

After B hits the floor, A continues to slide along the table. Given that — and that A comes to rest at P,

find d in terms of h.

Describe what would happen if