Two particles A and B, of mass 2 kg and 3 kg respectively, are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately before the collision the speed of A is 5 m s–1 and the speed of B is 6 m s–1. The magnitude of the impulse exerted on B by A is 14 N s. Find
the speed of A immediately after the collision,
(3)
the speed of B immediately after the collision.
(3)
A particle of weight 8 N is attached at C to the ends of two light inextensible strings AC and BC. The other ends, A and B, are attached to a fixed horizontal ceiling. The particle hangs at rest in equilibrium, with the strings in a vertical plane. The string AC is inclined at 35° to the horizontal and the string BC is inclined at 25° to the horizontal, as shown in Figure 1. Find
the tension in the string AC,
the tension in the string BC.
(8)
A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley P is fixed at the top of the plane. Two particles A and B, of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley P. The part of the string from A to P is parallel to a line of greatest slope of the plane and B hangs freely below P, as shown in Figure 2. The coefficient of friction between A and the plane is 
. Initially A is held at rest on the plane. The particles are released from rest with the string taut and A moves up the plane. Find the tension in the string immediately after the particles are released.
(9)
At time t = 0, two balls A and B are projected vertically upwards. The ball A is projected vertically upwards with speed 2 m s–1 from a point 50 m above the horizontal ground. The ball B is projected vertically upwards from the ground with speed 20 m s–1. At time t = T seconds, the two balls are at the same vertical height, h metres, above the ground. The balls are modelled as particles moving freely under gravity. Find
A particle P of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points A and B, where AB = 10 m, as shown in Figure 3. The speed of P at A is 2 m s–1. The particle P takes 3.5 s to move from A to B. Find
the coefficient of friction between P and the plane.
(5)
[ In this question i and j are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin O. ] A ship S is moving with constant velocity (3i + 3j) km h–1. At time t = 0, the position vector of S is (–4 i + 2 j) km.
Find the position vector of S at time t hours.
(2)
A ship T is moving with constant velocity (–2i + nj) km h–1. At time t = 0, the position vector of T is (6 i + j) km. The two ships meet at the point P.
Find the value of n.
(5)
A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle 
to the road, as shown in Figure 4. The vehicles are travelling at 20 m s–1 as they enter a zone where the speed limit is 14 m s–1. The truck’s brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s–1 is 100 m.
Find the deceleration of the truck and the car.
(3)
The constant braking force on the truck has magnitude R newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos 
= 0.9, find
the force in the towbar,
(4)
A uniform rod AB has length 2 m and mass 50 kg. The rod is in equilibrium in a horizontal position, resting on two smooth supports at C and D, where AC = 0.2 metres and DB = x metres, as shown in Figure 5. Given that the magnitude of the reaction on the rod at D is twice the magnitude of the reaction on the rod at C,
The support at D is now moved to the point E on the rod, where EB = 0.4 metres. A particle of mass m kg is placed on the rod at B, and the rod remains in equilibrium in a horizontal position. Given that the magnitude of the reaction on the rod at E is four times the magnitude of the reaction on the rod at C,
find the value of m.
(7)
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