Edexcel A Level Maths M1 · Mechanics 1 June 2012 question paper 6677

Two particles A and B, of mass 5 m kg and 2 m kg respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of A and B are 3 m s^–1 and 4 m s^–1 respectively. The direction of motion of A is unchanged by the collision. Immediately after the collision, the speed of A is 0.8 m s^–1.

Find the speed of B immediately after the collision.

In the collision, the magnitude of the impulse exerted on A by B is 3.3 N s.

Find the value of m.

A non-uniform rod AB has length 3 m and mass 4.5 kg. The rod rests in equilibrium, in a horizontal position, on two smooth supports at P and at Q, where AP = 0.8 m and QB = 0.6 m, as shown in Figure 1. The centre of mass of the rod is at G. Given that the magnitude of the reaction of the support at P on the rod is twice the magnitude of the reaction of the support at Q on the rod, find

the magnitude of the reaction of the support at Q on the rod,

the distance AG.

A box of mass 5 kg lies on a rough plane inclined at 30° to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N, as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find

the magnitude of the normal reaction of the plane on the box,

the coefficient of friction between the box and the plane.

A car is moving on a straight horizontal road. At time t = 0, the car is moving with speed 20 m s^–1 and is at the point A. The car maintains the speed of 20 m s^–1 for 25 s. The car

8 m s^–1. The car then moves with constant speed 8 m s^–1 for 60 s. The car then moves with constant acceleration until it is moving with speed 20 m s^–1 at the point B.

Sketch a speed-time graph to represent the motion of the car from A to B.

Find the time for which the car is decelerating.

Given that the distance from A to B is 1960 m,

find the time taken for the car to move from A to B.

A particle P is projected vertically upwards from a point A with speed u m s^–1. The point A is 17.5 m above horizontal ground. The particle P moves freely under gravity until it reaches the ground with speed 28 m s^–1.

Show that u = 21

At time t seconds after projection, P is 19 m above A.

Find the possible values of t.

The ground is soft and, after P reaches the ground, P sinks vertically downwards into the ground before coming to rest. The mass of P is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on P.

Find the vertical distance that P sinks into the ground before coming to rest.

[ In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin. ] A ship S is moving with constant velocity (–12i + 7.5j) km h^–1.

Find the direction in which S is moving, giving your answer as a bearing.

At time t hours after noon, the position vector of S is s km. When t = 0, s = 40 i – 6 j.

Write down s in terms of t.

A fixed beacon B is at the point with position vector (7 i + 12.5 j) km.

Find the distance of S from B when t = 3

Find the distance of S from B when S is due north of B.

Two particles P and Q, of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time t = 0, a constant force F of magnitude 4 N is applied to Q in the direction PQ, as shown in Figure 3. The system moves under the action of this force until t = 6 s. During the motion, the resistance to the motion of P has constant magnitude 1 N and the resistance to the motion of Q has constant magnitude 2 N. Find

the acceleration of the particles as the system moves under the action of F,

the speed of the particles at t = 6 s,

the tension in the rod as the system moves under the action of F.

At t = 6 s, F is removed and the system decelerates to rest. The resistances to motion are unchanged. Find

the distance moved by P as the system decelerates,

the thrust in the rod as the system decelerates.