Two moving objects interception (non-projectile)

Two objects (ships, aircraft) moving at constant velocities must intercept or find closest approach, using relative velocity methods rather than projectile equations under gravity.

4 questions · Standard +0.9

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Pre-U Pre-U 9795/2 2010 June Q2
9 marks Standard +0.8
2 At 1200 hours an aircraft, \(A\), sets out to intercept a second aircraft, \(B\), which is 200 km away on a bearing of \(300 ^ { \circ }\) and is flying due east at \(600 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Both aircraft are at the same altitude and continue to fly horizontally.
  1. (a) Find the bearing on which \(A\) should fly when travelling at \(800 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
    (b) Find the time at which \(A\) intercepts \(B\) in this case.
  2. Find the least steady speed at which \(A\) can fly to intercept \(B\).
Pre-U Pre-U 9795/2 2013 June Q10
6 marks Standard +0.8
10 Ship \(A\) is 15 km due south of ship \(B\). Ship \(B\) is travelling at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). Ship \(A\) is travelling at \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find
  1. the bearing, to the nearest degree, that \(A\) must take in order to get as close as possible to \(B\), [4]
  2. the time, in minutes, that it takes for the ships to be as close as possible.
Pre-U Pre-U 9795/2 2015 June Q11
11 marks Challenging +1.2
11 In a training exercise, a submarine is travelling due north at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The submarine commander sees his target 5 km away on a bearing of \(310 ^ { \circ }\). The target is travelling due east at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  1. If each of the submarine and target maintains its present course and speed, find the shortest distance between them.
  2. In fact, as soon as he sees the target, the submarine commander changes course, without changing speed, so as to intercept the target as quickly as possible. Find
    1. the course, in degrees, set by the submarine commander,
    2. the time taken, in minutes, to intercept the target from the moment that the course changes.
Pre-U Pre-U 9795/2 Specimen Q1
2 marks Standard +0.8
1 A ship \(A\) is steaming north at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Initially a ship \(B\) is at a distance 8 km due west of \(A\), and is steaming on a course such that it will take up a position 8 km directly ahead of \(A\) as quickly as possible.
  1. Given that the maximum speed of B is \(35 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), show that the bearing of this course is \(021 ^ { \circ }\), correct to the nearest degree.
  2. Find the distance that \(A\) moves between the instants when \(B\) is due west of \(A\) and when \(B\) is due north of \(A\), giving your answer to the nearest kilometre.