Closest approach of two objects

8. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] A ship \(A\) moves with constant velocity \(( 3 \mathbf { i } - 10 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) At time \(t\) hours, the position vector of \(A\) is \(\mathbf { r } \mathrm { km }\).
At time \(t = 0 , A\) is at the point with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\).

1. Find \(\mathbf { r }\) in terms of \(t\). Another ship \(B\) moves with constant velocity \(( 15 \mathbf { i } + 14 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At time \(t = 0 , B\) is at the point with position vector \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\).
2. Show that, at time \(t\) hours, $$\overrightarrow { A B } = [ ( 12 t - 10 ) \mathbf { i } + ( 24 t - 10 ) \mathbf { j } ] \mathrm { km }$$ Given that the two ships do not change course,
3. find the shortest distance between the two ships,
4. find the bearing of ship \(B\) from ship \(A\) when the ships are closest.
   
   \includegraphics[max width=\textwidth, alt={}]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-28_2820_1967_102_100}

5. Two dogs, Fido and Growler, are playing in a field. Fido is moving in a straight line so that at time \(t\) his position vector relative to a fixed origin, \(O\), is given by \([ ( 2 t - 3 ) \mathbf { i } + t \mathbf { j } ]\) metres. Growler is stationary at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } )\) metres, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.

1. Find the displacement vector of Fido from Growler in terms of \(t\).
2. Find the value of \(t\) for which the two dogs are closest.
3. Find the minimum distance between the two dogs.

4 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
Two cyclists, Aazar and Ben, are cycling on straight horizontal roads with constant velocities of \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and \(( 12 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) respectively. Initially, Aazar and Ben have position vectors \(( 5 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) and \(( 18 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) respectively, relative to a fixed origin.

1. Find, as a vector in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of Ben relative to Aazar.
2. The position vector of Ben relative to Aazar at time \(t\) hours after they start is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 13 + 6 t ) \mathbf { i } + ( 6 - 20 t ) \mathbf { j }$$
3. Find the value of \(t\) when Aazar and Ben are closest together.
4. Find the closest distance between Aazar and Ben.

4 Two boats, \(A\) and \(B\), are moving on straight courses with constant speeds. At noon, \(A\) and \(B\) have position vectors \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }\) and \(( - \mathbf { i } + \mathbf { j } ) \mathrm { km }\) respectively relative to a lighthouse. Thirty minutes later, the position vectors of \(A\) and \(B\) are ( \(- \mathbf { i } + 3 \mathbf { j }\) ) km and \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) respectively relative to the lighthouse.

1. Find the velocity of \(A\) relative to \(B\) in the form \(( m \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\), where \(m\) and \(n\) are integers.
2. The position vector of \(A\) relative to \(B\) at time \(t\) hours after noon is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 2 - 10 t ) \mathbf { i } + ( 1 + 6 t ) \mathbf { j }$$
3. Determine the value of \(t\) when \(A\) and \(B\) are closest together.
4. Find the shortest distance between \(A\) and \(B\).

1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]

A coastguard patrol boat \(C\) is moving with constant velocity \(( 8 \mathbf { i } + u \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Another ship \(S\) is moving with constant velocity \(( 12 \mathbf { i } + 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

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 \(S\),
2. 1. find the value of \(u\).
   2. Using this value of \(u\), find the time at which \(C\) would intercept \(S\). If instead, at noon, \(C\) is moving with velocity \(( 8 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and continues at this constant velocity,
3. find the distance of closest approach of \(C\) to \(S\).

1. 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 \(( 16 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m }\) and \(( - 5 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\) respectively. The velocity of \(P\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

Find the shortest distance between \(P\) and \(Q\) in the subsequent motion.

7 At midnight, ship \(A\) is 70 km due north of ship \(B\). Ship \(A\) travels with constant velocity \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(140 ^ { \circ }\). Ship \(B\) travels with constant velocity \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\).

1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\).
2. Find the distance between the ships when they are at their closest, and find the time when this occurs.

5 Two aircraft \(A\) and \(B\) are flying horizontally at the same height. \(A\) has constant velocity \(240 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\), and \(B\) has constant velocity \(185 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(310 ^ { \circ }\).

1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\). Initially \(A\) is 4500 m due west of \(B\). For the instant during the subsequent motion when \(A\) and \(B\) are closest together, find
2. the distance between \(A\) and \(B\),
3. the bearing of \(A\) from \(B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-2_823_650_1318_751} A boat \(A\) is travelling with constant speed \(6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(075 ^ { \circ }\). Boat \(B\) is travelling with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(025 ^ { \circ }\). At one instant, \(A\) is 2500 m due north of \(B\) (see diagram).

1. Find the magnitude and bearing of the velocity of \(A\) relative to \(B\).
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.

6 Two ships \(P\) and \(Q\) are moving on straight courses with constant speeds. At one instant \(Q\) is 80 km from \(P\) on a bearing of \(220 ^ { \circ }\). Three hours later, \(Q\) is 36 km due south of \(P\).

1. Show that the velocity of \(Q\) relative to \(P\) is \(19.1 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(063.8 ^ { \circ }\) (both correct to 3 significant figures).
2. Find the shortest distance between the two ships in the subsequent motion. Given that the speed of \(P\) is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and \(Q\) is travelling in the direction with bearing \(105 ^ { \circ }\), find
3. the bearing of the direction in which \(P\) is travelling,
4. the speed of \(Q\).

9 At noon a vessel, \(A\), leaves a port, \(O\), and travels at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(042 ^ { \circ }\). Also at noon a second vessel, \(B\), leaves another port, \(P , 13 \mathrm {~km}\) due north of \(O\), and travels at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(090 ^ { \circ }\). Take \(O\) as the origin and \(\mathbf { i }\) and \(\mathbf { j }\) as unit vectors east and north respectively.

1. Express the velocity vector of \(A\) relative to \(B\) in the form \(a \mathbf { i } + b \mathbf { j }\), where \(a\) and \(b\) are constants to be determined.
2. Express the position vector of \(A\) relative to \(B\), at time \(t\) hours after the vessels have left port, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
3. Explain why the scalar product of the vectors in parts (i) and (ii) is zero when the two vessels are closest together.
4. Find the time at which the two vessels are closest together. \(10 A\) and \(B\) are two points 6 m apart on a smooth horizontal surface. A particle, \(P\), of mass 0.5 kg is attached to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 20 N , and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 10 N , such that \(P\) is between \(A\) and \(B\).
5. Find the length \(A P\) when \(P\) is in equilibrium. \(P\) is held at the point \(C\), where \(C\) is between \(A\) and \(B\) and \(A C = 4.5 \mathrm {~m} . P\) is then released from rest. At time \(t\) seconds after being released, the displacement of \(P\) from the equilibrium position is \(y\) metres.
6. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 40 y$$
7. Find the time taken for \(P\) to reach the mid-point of \(A B\) for the first time. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-6_750_1187_258_479} Two particles, \(P\) and \(Q\), are projected simultaneously from the same point on a plane inclined at \(\alpha\) to the horizontal. \(P\) is projected up the plane and \(Q\) down the plane. Each particle is projected with speed \(V\) at an angle \(\theta\) to the plane. Both particles move in a vertical plane containing a line of greatest slope of the inclined plane and you are given that \(\alpha + \theta < \frac { 1 } { 2 } \pi\) (see diagram).
8. Show that the range of \(P\), up the plane, is given by $$\frac { 2 V ^ { 2 } \sin \theta } { g \cos ^ { 2 } \alpha } ( \cos \theta \cos \alpha - \sin \theta \sin \alpha ) .$$
9. Write down a similar expression for the range of \(Q\), down the plane.
10. Given that the range up the plane is a quarter of the range down the plane and that \(\alpha = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\), find \(\theta\).

Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \((20\mathbf{i} + 10\mathbf{j})\) km relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \((14\mathbf{i} - 6\mathbf{j})\) km. Three hours later, \(P\) is at the point with position vector \((29\mathbf{i} + 34\mathbf{j})\) km. The ship \(Q\) travels with velocity \(12\mathbf{j}\) km h\(^{-1}\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively. Find

1. the velocity of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. expressions for \(\mathbf{p}\) and \(\mathbf{q}\), in terms of \(t\), \(\mathbf{i}\) and \(\mathbf{j}\). [4]

At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d\) km.

1. By finding an expression for \(\overrightarrow{PQ}\), show that $$d^2 = 25t^2 - 92t + 292.$$ [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,

1. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer. [5]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \((1.2\mathbf{i} - 0.9\mathbf{j})\) m s\(^{-1}\).

1. Find the speed of \(H\). [2]

\includegraphics{figure_3} A horizontal field \(OABC\) is rectangular with \(OA\) due east and \(OC\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100\mathbf{j}\) m, relative to the fixed origin \(O\).

1. Write down the position vector of \(H\) at time \(t\) seconds after noon. [2]

At noon, another hiker \(K\) is at the point with position vector \((9\mathbf{i} + 46\mathbf{j})\) m. Hiker \(K\) is moving with constant velocity \((0.75\mathbf{i} + 1.8\mathbf{j})\) m s\(^{-1}\).

1. Show that, at time \(t\) seconds after noon, $$\overrightarrow{HK} = [(9 - 0.45t)\mathbf{i} + (2.7t - 54)\mathbf{j}] \text{ metres.}$$ [4]

Hence,

1. show that the two hikers meet and find the position vector of the point where they meet. [5]

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. The point \(A\) has position vector \(6\mathbf{j}\) m relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity \((5\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4\mathbf{i}\) ms\(^{-1}\).

1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds. [3 marks]
2. Show that the distance \(d\) m between \(P\) and \(Q\) at time \(t\) seconds is such that $$d^2 = 5t^2 - 24t + 36.$$ [5 marks]
3. Find the value of \(t\) for which \(d^2\) is a minimum. [3 marks]
4. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together. [4 marks]

Two ramblers, Alison and Bill, are out walking. At midday, Alison is at the fixed origin \(O\), and Bill is at the point with position vector \((-5\mathbf{i} + 12\mathbf{j})\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors. They are both walking with constant velocity – Alison at \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\), and Bill at a speed of \(2\sqrt{10}\) km h\(^{-1}\) in a direction parallel to the vector \((3\mathbf{i} + \mathbf{j})\).

1. Find the distance between the two ramblers at midday. [2 marks]
2. Show that the velocity of Bill is \((6\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). [3 marks]
3. Show that, at time \(t\) hours after midday, the position vector of Bill relative to Alison is $$[(4t - 5)\mathbf{i} + (12 - 3t)\mathbf{j}] \text{ km.}$$ [5 marks]
4. Show that the distance, \(d\) km, between the two ramblers is given by $$d^2 = 25t^2 - 112t + 169.$$ [2 marks]
5. Using your answer to part \((d)\), find the length of time to the nearest minute for which the distance between the Alison and Bill is less than 11 km. [5 marks]

Two ships \(A\) and \(B\) are sailing in the same direction at constant speeds of 12 km h\(^{-1}\) and 16 km h\(^{-1}\) respectively. They are sailing along parallel lines which are 4 km apart. When the distance between the ships is 4 km, \(B\) turns through 30° towards \(A\). Find the shortest distance between the ships in the subsequent motion. [7]

Two ships \(P\) and \(Q\) are moving with constant velocity. At 3 p.m., \(P\) is 20 km due north of \(Q\) and is moving at 16 km h\(^{-1}\) due west. To an observer on ship \(P\), ship \(Q\) appears to be moving on a bearing of \(030°\) at 10 km h\(^{-1}\). Find

1. 1. the speed of \(Q\),
   2. the direction in which \(Q\) is moving, giving your answer as a bearing to the nearest degree,
   [6]
2. the shortest distance between the ships, [3]
3. the time at which the two ships are closest together. [3]

A cyclist \(C\) is moving with a constant speed of \(10\) m s\(^{-1}\) due south. Cyclist \(D\) is moving with a constant speed of \(16\) m s\(^{-1}\) on a bearing of \(240°\).

1. Show that the magnitude of the velocity of \(C\) relative to \(D\) is \(14\) m s\(^{-1}\). [3]

At \(2\) pm, \(D\) is \(4\) km due east of \(C\).

1. Find
   1. the shortest distance between \(C\) and \(D\) during the subsequent motion,
   2. the time, to the nearest minute, at which this shortest distance occurs.
   [7]

Two horizontal roads cross at right angles. One is directed from south to north, and the other from east to west. A tractor travels north on the first road at a constant speed of 6 m s\(^{-1}\) and at noon is 200 m south of the junction. A car heads west on the second road at a constant speed of 24 m s\(^{-1}\) and at noon is 960 m east of the junction.

1. Find the magnitude and direction of the velocity of the car relative to the tractor. [6]
2. Find the shortest distance between the car and the tractor. [8]

\includegraphics{figure_6} A ship \(P\) is moving with constant velocity 7 m s\(^{-1}\) in the direction with bearing 110°. A second ship \(Q\) is moving with constant speed 10 m s\(^{-1}\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of 050° (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m.

1. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing 036°, to the nearest degree, and find the bearing of the other possible direction of this relative velocity. [3]

Given that the velocity of \(Q\) relative to \(P\) has bearing 036°, find

1. the bearing of the direction in which \(Q\) is moving, [4]
2. the magnitude of the velocity of \(Q\) relative to \(P\), [2]
3. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together, [3]
4. the bearing of \(P\) from \(Q\) when the two ships are closest together. [1]

\includegraphics{figure_2} Boat \(A\) is travelling with constant speed 7.9 m s\(^{-1}\) on a course with bearing 035°. Boat \(B\) is travelling with constant speed 10.5 m s\(^{-1}\) on a course with bearing 330°. At one instant, the boats are 1500 m apart with \(B\) on a bearing of 125° from \(A\) (see diagram).

1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\). [5]
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion. [2]
3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach. [2]

Two particles \(A\) and \(B\) have position vectors \(\mathbf{r}_A\) metres and \(\mathbf{r}_B\) metres at time \(t\) seconds, where $$\mathbf{r}_A = t^2\mathbf{i} + (3t - 1)\mathbf{j} \quad \text{and} \quad \mathbf{r}_B = (1 - 2t^2)\mathbf{i} + (3t - 2t^2)\mathbf{j}, \quad \text{for } t \geqslant 0.$$

1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed. [5]
2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d^2 = 13t^4 - 10t^2 + 2.$$ [3]
3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion. [6]