Vector部分真题集
2012年3月
1. [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 + 5explicitlyj) km h–1.
(a)Given that the two ships maintain the 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)
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午饭英文(d) find the time at which B will be due north of A.
(2)
(Total 13 marks)
2. A particle P moves with constant acceleration (2i – 3j) m s-2. At time t conds, 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)
broadcaster(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)
3. 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 highlight是什么意思j.
(2)
(b) the magnitude of F.
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(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 conds after the force has been removed, P is at the point B.
(c) Calculate the distance of B from O.
(6)
(Total 12 marks)
4. [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 + 5短裙英语j) 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)
5. A small boat S, drifting in the a, is modelled as a particle moving in a straight line at constant speed. When first sighted at 0900, S is at a point with position vector (4i – 6j) km relative to a fixed origin O, where i and j are unit vectors due east and due north respectively. At 0945, S is at the point with position vector (7i – 7.5j) km. At time t hours after 0900, S is at the point with position vector s km.
(a) Calculate the bearing on which S is drifting.
(4)
(b) Find an expression for s in terms of t.
(3)
At 1000 a motor boat M leaves O and travels with constant velocity (pi + qj) km h–1. Given that M intercepts S at 1015,
(c) calculate the value of p and the value of wereq.
(6)
(Total 13 marks)
6. A particle P moves in a horizontal plane. The acceleration of P is (–i + 2j) m s–2various. 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 conds, the velocity of Pzlt 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)
7. 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,
