Particles P and Q move in a plane with constant velocities. At time t = 0 the position vectors of P and Q, relative to a fixed point O in the plane, are (16i 
j) m and 
i + 4j) m respectively. The velocity of P is (i + 2j) m s
and the velocity of Q is (2i + j) m s
Find the shortest distance between P and Q in the subsequent motion.
(7)
When a woman walks due North at a constant speed of 4 km h
, the wind appears to be blowing from due East. When she runs due South at a constant speed of 8 km h
, the speed of the wind appears to be 20 km h
. Assuming that the velocity of the wind relative to the earth is constant, find
the speed of the wind,
the direction from which the wind is blowing.
(6)
Two smooth uniform spheres A and B with equal radii have masses m and 2 m respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, A has speed 3 u with its direction of motion at an angle θ to the line of centres, and B has speed u with its direction of motion at an angle θ to the line of centres, as shown in Figure 1. The coefficient of restitution between
Immediately after the collision, the speed of A is twice the speed of B. Find the size of the angle 
(12)
A car of mass 900 kg is moving along a straight horizontal road with the engine of the car working at a constant rate of 22.5 kW. At time t seconds, the speed of the car is v m s
(0 v 30) and the total resistance to the motion of the car has magnitude 25 v newtons.
Show that when the speed of the car is v m sí1, the acceleration of the car is
(3)
The time taken for the car to accelerate from 10 m sí1 to 20 m sí1 is T seconds.
Show that
(5)
Find the distance travelled by the car as it accelerates from 10 m s
to 20 m s
(6)
A particle P of mass 1.5 kg is attached to the midpoint of a light elastic spring AB, of natural length 2 m and modulus of elasticity 12 N. The end A of the spring is attached to a fixed point on a smooth horizontal floor. The end B is held at a point on the floor where AB = 6m. At time t = 0, P is at rest on the floor at the point O, where AO = 3m, as shown in Figure 2. The end B is now moved along the floor in such a way that AOB remains a straight line and at time t seconds, t ≥ 0,
At time t seconds, AP 
3 
x
m.
The general solution of this differential equation is
where C and D are constants.
Find the time at which P first comes to instantaneous rest.
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(5)
A smooth wire, with ends A and B, is in the shape of a semicircle of radius r. The line AB is horizontal and the midpoint of AB is O. The wire is fixed in a vertical plane. A small ring R of mass 2m is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at A and is attached to a
Show that the potential energy of the system is
(6)
Find the values of 
for which the system is in equilibrium.
(4)
Determine the stability of the position of equilibrium for which θ > 0
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(3)
Figure 4 represents the plan view of part of a smooth horizontal floor, where AB and BC are smooth vertical walls. The angle between AB and BC is 120°. A ball is projected along the floor towards AB with speed u m sí1 on a path at an angle of 60° to AB. The ball hits AB and then hits BC. The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is 
The speed of the ball immediately after it has hit BC is w m sí1
Find w in terms of u.
(7)
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