1.	[In this question, the unit vectors i and j are due east and due north respectively.]
	A particle P is moving with constant velocity (–5i + 8j) m s–1. Find
(a)	the speed of P,
(2)
 
(b)	the direction of motion of P, giving your answer as a bearing.
(3)
 
	At time t = 0, P is at the point A with position vector (7i – 10j) m relative to a fixed origin O. When t = 3 s, the velocity of P changes and it moves with velocity (ui + vj) m s–1, where u and v are constants. After a further 4 s, it passes through O and continues to move with velocity
(ui + vj) m s–1.
(c)	Find the values of u and v.
(5)
 
(d)	Find the total time taken for P to move from A to a position which is due south of A.
(3)
(Total 13 marks)

2.	A boat B is moving with constant velocity.  At noon, B is at the point with position vector 
(3i – 4j) km with respect to a fixed origin O. At 1430 on the same day, B is at the point with position vector (8i + 11j) km.
(a)	Find the velocity of B, giving your answer in the form pi + qj.
(3)
 
	At time t hours after noon, the position vector of B is b km.
(b)	Find, in terms of t, an expression for b.
(3)

	Another boat C is also moving with constant velocity.  The position vector of C, c km, at time t hours after noon, is given by
c = (–9i + 20j) + t(6i + λj),
	where λ is a constant. Given that C intercepts B,
(c)	find the value of λ,
(5)
 
(d)	show that, before C intercepts B, the boats are moving with the same speed.
(3)
(Total 14 marks)

3.	[In this question the unit vectors i and j are due east and north respectively.]
	A ship S is moving with constant velocity (–2.5i + 6j) km h–1. At time 1200, the position vector of S relative to a fixed origin O is (16i + 5j) km. Find
(a)	the speed of S,
(2)
(b)	the bearing on which S is moving.
(2)
 
	The ship is heading directly towards a submerged rock R. A radar tracking station calculates that, if S continues on the same course with the same speed, it will hit R at the time 1500.
(c)	Find the position vector of R.
(2)
 
	The tracking station warns the ship’s captain of the situation. The captain maintains S on its course with the same speed until the time is 1400. He then changes course so that S moves due north at a constant speed of 5 km h–1. Assuming that S continues to move with this new constant velocity, find
(d)	an expression for the position vector of the ship t hours after 1400,
(4)

(e)	the time when S will be due east of R,
(2)
(f)	the distance of S from R at the time 1600.
(3)
(Total 15 marks)

4.	[In this question the horizontal unit vectors i and j are due east and due north respectively.]
	A model boat A moves on a lake with constant velocity (–i + 6j) m s–1. At time t = 0, A is at the point with position vector (2i – 10j) m. Find
(a)	the speed of A,
(2)
(b)	the direction in which A is moving, giving your answer as a bearing.
(3)
 
	At time t = 0, a second boat B is at the point with position vector (–26i + 4j) m.
	Given that the velocity of B is (3i + 4j) m s–1,
(c)	show that A and B will collide at a point P and find the position vector of P.
(5)
	Given instead that B has speed 8 m s–1 and moves in the direction of the vector (3i + 4j),
(d)	find the distance of B from P when t = 7 s.
(6)
(Total 16 marks)

5.	[In this question, the unit vectors i and j are horizontal vectors due east and north respectively.]
	At time t = 0, a football player kicks a ball from the point A with position vector (2i + j) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity (5i + 8j) m s–1. Find
(a)	the speed of the ball,
(2)

(b)	the position vector of the ball after t seconds.
(2)
 
	The point B on the field has position vector (10i + 7j) m.
(c)	Find the time when the ball is due north of B.
(2)
 
	At time t = 0, another player starts running due north from B and moves with constant speed v m s–1. Given that he intercepts the ball,
(d)	find the value of v.
(6)
 
(e)	State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball’s motion to make the model more realistic.
(1)
(Total 13 marks)

6.	Two ships P and Q are travelling at night with constant velocities. At midnight, P is at the point with position vector (20i + 10j) km relative to a fixed origin O. At the same time, Q is at the point with position vector (14i – 6j) km. Three hours later, P is at the point with position vector (29i + 34j) km. The ship Q travels with velocity 12j km h–1. At time t hours after midnight, the position vectors of P and Q are p km and q km respectively. Find
(a)	the velocity of P, in terms of i and j,
(2)
 
(b)	expressions for p and q, in terms of t, i and j.
(4)
 
	At time t hours after midnight, the distance between P and Q is d km.
(c)	By finding an expression for , show that
d 2 = 25t2 – 92t + 292.
(5)

	Weather conditions are such that an observer on P can only see the lights on Q when the distance between P and Q is 15 km or less. Given that when t = 1, the lights on Q move into sight of the observer,
(d)	find the time, to the nearest minute, at which the lights on Q move out of sight of the observer.
(5)
(Total 16 marks)

7.	A particle P moves in a horizontal plane. The acceleration of P is (–i + 2j) m s–2. At time t = 0, the velocity of P is (2i – 3j) m s–1.
(a)	Find, to the nearest degree, the angle between the vector j and the direction of motion of P when t = 0.
(3)
 
	At time t seconds, the velocity of P is v m s–1. Find
(b)	an expression for v in terms of t, in the form ai + bj,
(2)
 
(c)	the speed of P when t = 3,
(3)
 
(d)	the time when P is moving parallel to i.
(2)
(Total 10 marks)

8.	[In this question the vectors i and j are horizontal unit vectors in the directions due east and due north respectively.]
	Two boats A and B are moving with constant velocities. Boat A moves with velocity 9j km h–1. Boat B moves with velocity (3i + 5j) km h–1.
 
(a)	Find the bearing on which B is moving.
(2)

	At noon, A is at point O, and B is 10 km due west of O. At time t hours after noon, the position vectors of A and B relative to O are a km and b km respectively. 
(b)	Find expressions for a and b in terms of t, giving your answer in the form pi + qj. 
(3)
 
(c)	Find the time when B is due south of A.
(2)
 
	At time t hours after noon, the distance between A and B is d km. By finding an expression for
,
 
(d)	show that d2 = 25t2 – 60t + 100.
(4)
 
	At noon, the boats are 10 km apart.
(e)	Find the time after noon at which the boats are again 10 km apart.
(3)
(Total 14 marks)

9.	A particle P of mass 3 kg is moving under the action of a constant force F newtons. At t = 0, P has velocity (3i – 5j) m s –1. At t = 4 s, the velocity of P is (–5i + 11j) m s–1. Find
(a)	the acceleration of P, in terms of i and j.
(2)
 
(b)	the magnitude of F.
(4)

	At t = 6 s, P is at the point A with position vector (6i – 29j) m relative to a fixed origin O. At this instant the force F newtons is removed and P then moves with constant velocity. Three seconds after the force has been removed, P is at the point B.
(c)	Calculate the distance of B from O.
(6)
(Total 12 marks)

10.	A particle P moves with constant acceleration (2i – 3j) m s-2.  At time t seconds, its velocity is v m s-1. When t = 0, v  = -2i + 7j.
(a)	Find the value of t when P is moving parallel to the vector i.
(4)
(b)	Find the speed of P when t = 3.
(3)
(c)	Find the angle between the vector j and the direction of motion of P when t = 3. 
(3)
(Total 10 marks)

11.	A particle P of mass 0.4 kg is moving under the action of a constant force F newtons. Initially the velocity of P is (6i – 27j) m s–1 and 4 s later the velocity of P is (-14i + 21j) m s–1.
(a)	Find, in terms of i and j, the acceleration of P.
(3)
(b)	Calculate the magnitude of F.
(3)
(Total 6 marks)

12.	Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (6i + 12j) km, where i and j are unit vectors directed due east and due north respectively. The ship P is moving with velocity 10j km h–1 and Q is moving with velocity (-8i + 6j) km h-1. At time t hours the position vectors of P and Q relative to O are p km and q km respectively.
(a)	Find p and q in terms of t.
(3)
(b)	Calculate the distance of Q from P when t = 3.
(3)
(c)	Calculate the value of t when Q is due north of P.
(2)
(Total 8 marks)

13.	[In this question, the horizontal unit vectors i and j are directed due East and North respectively.]
	A coastguard station O monitors the movements of ships in a channel. At noon, the station’s radar records two ships moving with constant speed. Ship A is at the point with position vector (–5i + 10j) km relative to O and has velocity (2i + 2j) km h–1. Ship B is at the point with position vector (3i  + 4j) km and has velocity (–2i + 5j) km h–1. 
 
(a)	Given that the two ships maintain these velocities, show that they collide.
(6)
	The coast guard radios ship A and orders it to reduce its speed to move with velocity (i + j) km h–1.
	Given that A obeys this order and maintains this new constant velocity,
(b)	find an expression for the vector at time t hours after noon.
(2)
 
(c)	find, to 3 significant figures, the distance between A and B at 1400 hours, 
(3)
(d)	find the time at which B will be due north of A.
(2)
(Total 13 marks)