1.10h Vectors in kinematics: uniform acceleration in vector form

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.

2 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. Two helicopters, \(A\) and \(B\), are flying with constant velocities of \(( 20 \mathbf { i } - 10 \mathbf { j } + 20 \mathbf { k } ) \mathrm { ms } ^ { - 1 }\) and \(( 30 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. At noon, the position vectors of \(A\) and \(B\) relative to a fixed origin, \(O\), are \(( 8000 \mathbf { i } + 1500 \mathbf { j } + 3000 \mathbf { k } ) \mathrm { m }\) and \(( 2000 \mathbf { i } + 500 \mathbf { j } + 1000 \mathbf { k } ) \mathrm { m }\) respectively.

1. Write down the velocity of \(A\) relative to \(B\).
2. Find the position vector of \(A\) relative to \(B\) at time \(t\) seconds after noon.
3. Find the value of \(t\) when \(A\) and \(B\) are closest together.

2 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
Two runners, Albina and Brian, are running on level parkland with constant velocities of \(( 5 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. Initially, the position vectors of Albina and Brian are \(( - 60 \mathbf { i } + 160 \mathbf { j } ) \mathrm { m }\) and \(( 40 \mathbf { i } - 90 \mathbf { j } ) \mathrm { m }\) respectively, relative to a fixed origin in the parkland.

1. Write down the velocity of Brian relative to Albina.
2. Find the position vector of Brian relative to Albina \(t\) seconds after they leave their initial positions.
3. Hence determine whether Albina and Brian will collide if they continue running with the same velocities.

4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed east, north and vertically upwards respectively. At time \(t = 0\), the position vectors of two small aeroplanes, \(A\) and \(B\), relative to a fixed origin \(O\) are \(( - 60 \mathbf { i } + 30 \mathbf { k } ) \mathrm { km }\) and \(( - 40 \mathbf { i } + 10 \mathbf { j } - 10 \mathbf { k } ) \mathrm { km }\) respectively. The aeroplane \(A\) is flying with constant velocity \(( 250 \mathbf { i } + 50 \mathbf { j } - 100 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the aeroplane \(B\) is flying with constant velocity \(( 200 \mathbf { i } + 25 \mathbf { j } + 50 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Write down the position vectors of \(A\) and \(B\) at time \(t\) hours.
2. Show that the position vector of \(A\) relative to \(B\) at time \(t\) hours is \(( ( - 20 + 50 t ) \mathbf { i } + ( - 10 + 25 t ) \mathbf { j } + ( 40 - 150 t ) \mathbf { k } ) \mathrm { km }\).
3. Show that \(A\) and \(B\) do not collide.
4. Find the value of \(t\) when \(A\) and \(B\) are closest together.
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-13_2484_1709_223_153}

4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. A helicopter, \(A\), is travelling in the direction of the vector \(- 2 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\) with constant speed \(140 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Another helicopter, \(B\), is travelling in the direction of the vector \(2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\) with constant speed \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Find the velocity of \(A\) relative to \(B\).
2. Initially, the position vectors of \(A\) and \(B\) are \(( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) and \(( - 3 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) respectively, relative to a fixed origin. Write down the position vector of \(A\) relative to \(B , t\) hours after they leave their initial positions.
3. Find the distance between \(A\) and \(B\) when they are closest together.
   
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-10_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-11_2486_1714_221_153}

6 A ship and a navy frigate are a distance of 8 km apart, with the frigate on a bearing of \(120 ^ { \circ }\) from the ship, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-16_451_549_411_760} The ship travels due east at a constant speed of \(50 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The frigate travels at a constant speed of \(35 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. 1. Find the bearings, to the nearest degree, of the two possible directions in which the frigate can travel to intercept the ship.
      [0pt] [5 marks]
   2. Hence find the shorter of the two possible times for the frigate to intercept the ship.
      [0pt] [5 marks]
2. The captain of the frigate would like the frigate to travel at less than \(35 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find the minimum speed at which the frigate can travel to intercept the ship.
   [0pt] [3 marks] \(7 \quad\) A particle is projected from a point \(O\) on a plane which is inclined at an angle \(\theta\) to the horizontal. The particle is projected up the plane with velocity \(u\) at an angle \(\alpha\) above the horizontal. The particle strikes the plane for the first time at a point \(A\). The motion of the particle is in a vertical plane which contains the line \(O A\). \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-20_469_624_502_685}
   1. Find, in terms of \(u , \theta , \alpha\) and \(g\), the time taken by the particle to travel from \(O\) to \(A\).
   2. The particle is moving horizontally when it strikes the plane at \(A\). By using the identity \(\sin ( P - Q ) = \sin P \cos Q - \cos P \sin Q\), or otherwise, show that $$\tan \alpha = k \tan \theta$$ where \(k\) is a constant to be determined.
      [0pt] [5 marks]
      \includegraphics[max width=\textwidth, alt={}]{bcd20c69-cace-408c-8961-169c19ff0231-24_2488_1728_219_141}

2. A particle \(P\) of mass 0.25 kg is moving on a horizontal plane. At time \(t\) seconds the velocity, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), of \(P\) relative to a fixed origin \(O\) is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane.

1. Find the acceleration of \(P\) in terms of \(t\).
2. Find, correct to 3 significant figures, the magnitude of the resultant force acting on \(P\) when \(t = 1\).
   (4 marks)

1. The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula

$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the acceleration of the toy in terms of \(t\).
2. Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
3. Explain why this model is unlikely to be realistic for large values of \(t\).

2. Ship \(A\) is steaming on a bearing of \(060 ^ { \circ }\) at \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and at 9 a.m. it is 20 km due west of a second ship \(B\). Ship \(B\) steams in a straight line.

1. Find the least speed of \(B\) if it is to intercept \(A\). Given that the speed of \(B\) is \(24 \mathrm {~km} \mathrm {~h} ^ { - 1 }\),
2. find the earliest time at which it can intercept \(A\).

5. At time \(t = 0\) particles \(P\) and \(Q\) start simultaneously from points which have position vectors \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). The velocities of \(P\) and \(Q\) are \(( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively.

1. Show that \(P\) and \(Q\) collide and find the position vector of the point at which they collide. A third particle \(R\) moves in such a way that its velocity relative to \(P\) is parallel to the vector ( \(- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) ) and its velocity relative to \(Q\) is parallel to the vector \(( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\). Given that all three particles collide simultaneously, find
2. 1. the velocity of \(R\),
   2. the position vector of \(R\) at time \(t = 0\).

2. \begin{figure}[h] \end{figure} A man, who rows at a speed \(v\) through still water, rows across a river which flows at a speed \(u\). The man sets off from the point \(A\) on one bank and wishes to land at the point \(B\) on the opposite bank, where \(A B\) is perpendicular to both banks, as shown in Fig. 1.

1. Show that, for this to be possible, \(v > u\). Given that \(v < u\) and that he rows from \(A\) so as to reach a point \(C\), on the opposite bank, which is as close to \(B\) as possible,
2. find, in terms of \(u\) and \(v\), the ratio of \(B C\) to the width of the river.
   (5)

3. A man walks due north at a constant speed \(u\) and the wind seems to him to be blowing from the direction \(30 ^ { \circ }\) east of north. On his return journey, when he is walking at the same speed \(u\) due south, the wind seems to him to be blowing from the direction \(30 ^ { \circ }\) south of east. Assuming that the velocity, \(\mathbf { w }\), of the wind relative to the earth is constant, find

1. the magnitude of \(\mathbf { w }\), in terms of \(u\),
2. the direction of \(\mathbf { w }\).

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

An aeroplane makes a journey from a point \(P\) to a point \(Q\) which is due east of \(P\). The wind velocity is \(w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )\), where \(w\) is a positive constant. The velocity of the aeroplane relative to the wind is \(v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )\), where \(v\) is a constant and \(v > w\). Given that \(\theta\) and \(\phi\) are both acute angles,

1. show that \(v \sin \phi = w \sin \theta\),
2. find, in terms of \(v , w\) and \(\theta\), the speed of the aeroplane relative to the ground.

3. At noon, two boats \(A\) and \(B\) are 6 km apart with \(A\) due east of \(B\). Boat \(B\) is moving due north at a constant speed of \(13 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(A\) is moving with constant speed \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and sets a course so as to pass as close as possible to boat \(B\). Find

1. the direction of motion of \(A\), giving your answer as a bearing,
2. the time when the boats are closest,
3. the shortest distance between the boats.

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

A ship \(A\) has constant velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) and a ship \(B\) has constant velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At noon, the position vectors of the ships \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) respectively. Find

1. the time at which the two ships are closest together,
2. the length of time for which ship \(A\) is within 2 km of ship \(B\).

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.

3 From a speedboat, a ship is sighted on a bearing of \(045 ^ { \circ }\). The ship has constant velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(120 ^ { \circ }\). The speedboat travels in a straight line with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and intercepts the ship.

1. Find the bearing of the course of the speedboat.
2. Find the magnitude of the velocity of the ship relative to the speedboat.

4 A cruise ship \(C\) is sailing due north at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat \(B\), initially 2000 m due west of \(C\), sails with constant speed \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight line course which takes it as close as possible to \(C\).

1. Find the bearing of the direction in which \(B\) sails.
2. Find the shortest distance between \(B\) and \(C\) in the subsequent motion.

4 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-2_364_1313_1224_376} An unidentified aircraft \(U\) is flying horizontally with constant velocity \(250 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(040 ^ { \circ }\). Two spotter planes \(P\) and \(Q\) are flying horizontally at the same height as \(U\), and at one instant \(P\) is 15000 m due west of \(U\), and \(Q\) is 15000 m due east of \(U\) (see diagram).

1. Plane \(P\) is flying with constant velocity \(210 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(070 ^ { \circ }\).

1 Alan is running in a straight line on a bearing of \(090 ^ { \circ }\) at a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\). Ben sees Alan when they are 50 m apart and Alan is on a bearing of \(060 ^ { \circ }\) from Ben. Ben sets off immediately to intercept Alan by running at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the bearing on which Ben should run to intercept Alan.
2. Calculate the magnitude of the velocity of Ben relative to Alan and find the time it takes, from the moment Ben sees Alan, for Ben to intercept Alan.

3 Two planes, \(A\) and \(B\), flying at the same altitude, are participating in an air show. Initially the planes are 400 m apart and plane \(B\) is on a bearing of \(130 ^ { \circ }\) from plane \(A\). Plane \(A\) is moving due south with a constant speed of \(75 \mathrm {~ms} ^ { - 1 }\). Plane \(B\) is moving at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) and has set a course to get as close as possible to \(A\).

1. Find the bearing of the course set by \(B\) and the shortest distance between the two planes in the subsequent motion.
2. Find the total distance travelled by \(A\) and \(B\) from the instant when they are initially 400 m apart to the point of their closest approach.

2 A ship \(S\) is travelling with constant speed \(5 \mathrm {~ms} ^ { - 1 }\) on a course with bearing \(325 ^ { \circ }\). A second ship \(T\) observes \(S\) when \(S\) is 9500 m from \(T\) on a bearing of \(060 ^ { \circ }\) from \(T\). Ship \(T\) sets off in pursuit, travelling with constant speed \(8.5 \mathrm {~ms} ^ { - 1 }\) in a straight line.

1. Find the bearing of the course which \(T\) should take in order to intercept \(S\).
2. Find the distance travelled by \(S\) from the moment that \(T\) sets off in pursuit until the point of interception.

3. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, relative to a fixed origin. The particle moves in such a way that $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ At \(t = 0 , P\) is moving with velocity ( \(8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the speed of \(P\) when \(t = \frac { 1 } { 2 } \ln 2\).

3 A particle Q of mass \(m\) moves in a horizontal plane under the action of a single force \(\mathbf { F }\). At time \(t , \mathrm { Q }\) has velocity \(\binom { 2 } { 3 t - 2 }\).

1. Find an expression for \(\mathbf { F }\) in terms of \(m\). At time \(t\), the displacement of Q is given by \(\mathbf { r } = \binom { x } { y }\). When \(t = 1 , \mathrm { Q }\) is at the point with position vector \(\binom { 4 } { - 4 }\).
2. Find the equation of the path of Q , giving your answer in the form \(y = a x ^ { 2 } + b x + c\), where \(a\), \(b\) and \(c\) are constants to be determined.
3. What can you deduce about the path of Q from the value of the constant \(c\) you found in part (b)?

6. A ball is projected with velocity \(( 4 w \mathbf { i } + 7 w \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the top of a vertical tower. After 5 seconds, the ball hits the ground at a point that is 60 m horizontally from the foot of the tower. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.

1. Find the value of \(w\) and hence determine the height of the tower.
2. Determine the proportion of the 5 seconds for which the ball is on its way down.