Relative motion

Ship B is moving due south at a constant speed of 6

When a man walks due West at a constant speed of 4 km h^–1, the wind appears to be blowing from due South. When he runs due North at a constant speed of 8 km h^–1, the speed of the wind appears to be 5 km h^–1. The velocity of the wind relative to the Earth is constant with magnitude w km h^–1. Find the two possible values of w.

A horizontal square field, PQRS, has sides of length 75 m. Ali is at corner P of the field and Beth is at corner Q of the field. Ali starts to walk in a straight line along the diagonal of the field from P to R at a constant speed of 1.5 m s^–1. Beth sees Ali start to walk, waits 10 seconds, and then walks from Q to intercept Ali. Beth walks in a straight line at a constant speed of 2 m s^–1. Find

the time from the instant Beth leaves Q until the instant that she intercepts Ali,

the direction Beth should take.

[ In this question the horizontal unit vectors i and j are due east and due north respectively.] A ship A has constant velocity (4i + 2j)kmh^–1 and a ship B has constant velocity (–i + 3j)km h^–1. At noon, the position vectors of the ships A and B with respect to a fixed origin O are (–2 i + j)km and (5 i – 2 j)km respectively. Find

the time at which the two ships are closest together,

the length of time for which ship A is within 2km of ship B.

A cyclist riding due north at a steady speed of 12 km h^–1 notices that the wind appears to come from the north-west. At the same time, another cyclist, moving on a bearing of 120° and also riding at a steady speed of 12 km h^–1, notices that the wind appears to come from due south. The velocity of the wind is assumed to be constant. Find

the wind speed,

the direction from which the wind is blowing, giving your answer as a bearing.

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 sand the velocity of Q is (2i + j) m s Find the shortest distance between P and Q in the subsequent motion.

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.

A particle A has constant velocity (3i + j) m s^–1 and a particle B has constant velocity (i – k) m s^–1. At time t = 0 seconds, the position vectors of the particles A and B with respect to a fixed origin O are (–6 i + 4 j – 3 k) m and (–2 i + 2 j + 3 k) m respectively.

Show that, in the subsequent motion, the minimum distance between A and B is 4 ¥ 2 m.

Find the position vector of A at the instant when the distance between A and B is a minimum.

A ship A is travelling at a constant speed of 30 km h^–1 on a bearing of 050°. Another ship B is travelling at a constant speed of v km h^–1 and sets a course to intercept A. At 1400 hours B is 20 km from A and the bearing of A from B is 290°.

Find the least possible value of v.

Given that v = 32,

find the time at which B intercepts A.

At noon two ships A and B are 20 km apart with A on a bearing of 230 from B. Ship B is moving at 6 km h^–1 on a bearing of 015. The maximum speed of A is 12 km h^–1. Ship A sets a course to intercept B as soon as possible.

Find the course set by A, giving your answer as a bearing to the nearest degree.

Find the time at which A intercepts B.

[ In this question i and j are horizontal unit vectors due east and due north respectively.] Boat A is moving with velocity (3i + 4j) km h^–1 and boat B is moving with velocity (6i – 5j) km h^–1. Find

the magnitude of the velocity of A relative to B,

the direction of the velocity of A relative to B, giving your answer as a bearing.

A river is 50 m wide and flows between two straight parallel banks. The river flows with a uniform speed of m s^–1 parallel to the banks. The points A and B are on opposite banks of the river and AB is perpendicular to both banks of the river, as shown in Figure 1. Keith and Ian decide to swim across the river. The speed relative to the water of both

Keith sets out from A and crosses the river in the least possible time, reaching the opposite bank at the point C. Find

the time taken by Keith to reach C,

the distance BC.

Ian sets out from A and swims in a straight line so as to land on the opposite bank at B.

Find the time taken by Ian to reach B.

At 10 a.m. two walkers A and B are 4 km apart with A due north of B. Walker A is moving due east at a constant speed of 6 km h^–1. Walker B is moving with constant speed 5 km h^–1 and walks in the straight line which allows him to pass as close as possible to A. Find

the direction of motion of B, giving your answer as a bearing,

the least distance between A and B,

the time when the distance between A and B is least.

A coastguard ship C is due south of a ship S. Ship S is moving at a constant speed of 12 km h^–1 on a bearing of 140 Ship C moves in a straight line with constant speed V km h^–1 in order to intercept S.

Find, giving your answer to 3 significant figures, the minimum possible value for V.

It is now given that V = 14

Find the bearing of the course that C takes to intercept S.

[ In this question the unit vectors i and j are due east and due north respectively.] A coastguard patrol boat C is moving with constant velocity (8i + uj) km h^–1. Another ship S is moving with constant velocity (12i + 16j) km h^–1.

Find, in terms of u, the velocity of C relative to S.

At noon, S is 10 km due west of C. If C is to intercept S,

find the value of u.

Using this value of u, find the time at which C would intercept S.

If instead, at noon, C is moving with velocity (8i + 8j) km h^–1 and continues at this constant velocity,

find the distance of closest approach of C to S.

A hiker walking due east at a steady speed of 5 km h^–1 notices that the wind appears to come from a direction with bearing 050. At the same time, another hiker moving on a bearing of 320, and also walking at 5 km h^–1, notices that the wind appears to come from due north. Find

the direction from which the wind is blowing,

the wind speed.

[ In this question i and j are unit vectors due east and due north respectively ] A man cycles at a constant speed u m s^–1 on level ground and finds that when his velocity is uj m s^–1 the velocity of the wind appears to be v(3i – 4j) m s^–1, where v is a positive constant.

Find, in terms of u, the true velocity of the wind.

At 12 noon, ship A is 8 km due west of ship B. Ship A is moving due north at a constant speed of 10 km h^–1. Ship B is moving at a constant speed of 6 km h^–1 on a bearing so that it passes as close to A as possible.

Find the bearing on which ship B moves.

Find the shortest distance between the two ships.

Find the time when the two ships are closest.

At noon a motorboat P is 2 km north-west of another motorboat Q. The motorboat P is moving due south at 20 m s^–1. The motorboat Q is pursuing motorboat P at a speed of 12 m s^–1 and sets a course in order to get as close to motorboat P as possible.

Find the course set by Q, giving your answer as a bearing to the nearest degree.

Find the shortest distance between P and Q.

Find the distance travelled by Q from its position at noon to the point of closest approach.

[ In this question i and j are unit vectors due east and due north respectively.] A ship P is moving with velocity (5i – 4j) km h^-1 and a ship Q is moving with velocity (3i + 7j) km h^-1. Find the direction that ship Q appears to be moving in, to an observer on ship P, giving your answer as a bearing.

A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time t = 0, a boat leaves a point O on one bank and moves in a straight line across the river to a point P on the opposite bank. Its path OP is perpendicular to both banks and OP = 30 m, as shown in Figure 2. The speed of flow of the river, r m s^-1, at a point on OP which is at a distance x m from O, is modelled as

The speed of the boat relative to the water is constant at 5 m s^-1. At time t seconds the boat is at a distance x m from O and is moving with speed v m s^-1 in the direction OP.

Show that

Hence show that

Find the total time taken for the boat to cross the river from O to P.