1.10h Vectors in kinematics: uniform acceleration in vector form

3. The position vectors \(\mathbf { r } _ { A }\) and \(\mathbf { r } _ { B }\), in kilometres, of two small aeroplanes \(A\) and \(B\) relative to a fixed point \(O\) are given by $$\begin{aligned} & \mathbf { r } _ { A } = ( 60 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) + ( 168 \mathbf { i } + 132 \mathbf { j } ) t \\ & \mathbf { r } _ { B } = ( 62 \mathbf { i } + 3 \mathbf { k } ) + ( 160 \mathbf { i } + p \mathbf { j } + q \mathbf { k } ) t \end{aligned}$$ where \(t\) denotes the time in hours after 9:00 a.m. and \(p , q\) are constants.
The aeroplanes \(A\) and \(B\) are on course to collide.

1. Show that \(p = 140\) and \(q = 4\).
2. Find an expression for the square of the distance between \(A\) and \(B\) at time \(t\) hours after 9:00 a.m.
3. Both aeroplanes have systems that activate an alarm if they come within 600 m of each other. Determine the time when the alarms are first activated.

8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).

1. Show that \(k = - 5\).
2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).

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

1. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
2. expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\). At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
3. By finding an expression for \(\overrightarrow { P Q }\), show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ 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,
4. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
   1. In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
   2. the acceleration of the aircraft,
   3. the distance \(B C\).
   4. Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
   5. the speed of \(A\) immediately after the collision,
   6. the magnitude of the impulse exerted on \(B\) in the collision.
      
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
   \end{figure}

2 A particle \(P\) moves with acceleration \(( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds.
2. Find the speed of \(P\) when \(t = 0.5\).

5 An aeroplane flies in air that is moving due east at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the aeroplane relative to the air is \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The aeroplane actually travels on a bearing of \(030 ^ { \circ }\).

1. Show that \(V = 86.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the magnitude of the resultant velocity of the aeroplane.

5 Tom is on a fairground ride.
Tom's position vector, \(\mathbf { r }\) metres, at time \(t\) seconds is given by $$\mathbf { r } = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j } + ( 10 - 0.4 t ) \mathbf { k }$$ The perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane and the unit vector \(\mathbf { k }\) is directed vertically upwards.

1. 1. Find Tom's position vector when \(t = 0\).
   2. Find Tom's position vector when \(t = 2 \pi\).
   3. Write down the first two values of \(t\) for which Tom is directly below his starting point.
2. Find an expression for Tom's velocity at time \(t\).
3. Tom has mass 25 kg . Show that the resultant force acting on Tom during the motion has constant magnitude. State the magnitude of the resultant force.
   (5 marks)

3 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector of the particle is \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( 2 \mathrm { e } ^ { \frac { 1 } { 2 } t } - 8 t + 5 \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the speed of the particle when \(t = 3\).
   2. State the direction in which the particle is travelling when \(t = 3\).
3. Find the acceleration of the particle when \(t = 3\).
4. The mass of the particle is 7 kg . Find the magnitude of the resultant force on the particle when \(t = 3\).

3 A fishing boat is travelling between two ports, \(A\) and \(B\), on the shore of a lake. The bearing of \(B\) from \(A\) is \(130 ^ { \circ }\). The fishing boat leaves \(A\) and travels directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A patrol boat on the lake is travelling with speed \(4 \mathrm {~ms} ^ { - 1 }\) on a bearing of \(040 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}

1. Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
2. When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
   1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
   2. Given that the patrol boat intercepts the fishing boat before it reaches \(B\), find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
   3. State a modelling assumption necessary for answering this question, other than the boats being particles.

19

1. (ii) Verify that \(k = 0.8\) [0pt] [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 19
2. Find the position vector of Amba when \(t = 4\) [0pt] [3 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 19
3. At both \(t = 0\) and \(t = 4\) there is a distance of 5 metres between Jo and Amba's positions. Determine the shortest distance between their two parallel lines of motion.
   Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-32_2492_1721_217_150}

16 Two particles, \(P\) and \(Q\), move in the same horizontal plane. Particle \(P\) is initially at rest at the point with position vector \(( - 4 \mathbf { i } + 5 \mathbf { j } )\) metres and moves with constant acceleration \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\) Particle \(Q\) moves in a straight line, passing through the points with position vectors \(( \mathbf { i } - \mathbf { j } )\) metres and \(( 10 \mathbf { i } + c \mathbf { j } )\) metres. \(P\) and \(Q\) are moving along parallel paths.
16

1. Show that \(c = - 13\) 16
2. (i) Find an expression for the position vector of \(P\) at time \(t\) seconds.
   16 (b) (ii) Hence, prove that the paths of \(P\) and \(Q\) are not collinear.

17 A particle is moving such that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, is given by $$\mathbf { r } = \mathrm { e } ^ { t } \cos t \mathbf { i } + \mathrm { e } ^ { t } \sin t \mathbf { j }$$ Show that the magnitude of the acceleration of the particle, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$a = 2 \mathrm { e } ^ { t }$$ Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-27_2490_1728_217_141}

6 A particle, of mass 5 kg , moves on a circular path so that at time \(t\) seconds it has position vector \(\mathbf { r }\) metres, where $$\mathbf { r } = ( 2 \sin 3 t ) \mathbf { i } + ( 2 \cos 3 t ) \mathbf { j }$$ 6

1. Prove that the velocity of the particle is perpendicular to its position vector.
   6
2. Prove that the magnitude of the resultant force on the particle is constant.

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors and position vectors are given relative to a fixed origin.]

A ship \(A\) is moving with constant velocity.
At 1 pm , the position vector of \(A\) is \(( 25 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\).
At 3 pm , the position vector of \(A\) is \(( 55 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\).
At time \(t\) hours after 1 pm , the position vector of \(A\) is \(\mathbf { r } _ { A } \mathrm {~km}\).

1. Show that \(\mathbf { r } _ { A } = ( 25 + 15 t ) \mathbf { i } + ( 10 + 12 t ) \mathbf { j }\) The speed of \(A\) is \(V \mathrm {~ms} ^ { - 1 }\)
2. Find the value of \(V\). A ship \(B\) is moving with constant velocity \(( 20 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At 1 pm , the position vector of \(B\) is \(( 35 \mathbf { i } + 51 \mathbf { j } ) \mathrm { km }\).
   At 2:30 pm, \(B\) passes through the point \(P\).
3. Show that \(A\) also passes through \(P\).

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 \(B\).

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 \(\theta\).

8 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_757_729_260_708} An aircraft carrier, \(A\), is heading due north at \(40 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A destroyer, \(D\), which is 8 km south-west of \(A\), is ordered to take up a position 3 km east of \(A\) as quickly as possible. The speed of \(D\) is \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) (see diagram). Find the bearing, \(\theta\), of the course that \(D\) should take, giving your answer to the nearest degree.

7 At a given instant two stunt cars, \(X\) and \(Y\), are at distances 500 m and 800 m respectively from the point of intersection, \(O\), of two straight roads crossing at right angles. The stunt cars are approaching \(O\) at uniform speeds of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, one on each road. Find, in either order,

1. the time taken to reach the point of closest approach,
2. the shortest distance between the stunt cars.

The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).

1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]

A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.

1. Find the velocity of the particle when \(t = 4\). [4]
2. Find the kinetic energy of the particle when \(t = 4\). [2]
3. Find the distance of the particle from the origin when \(t = 2\). [6]

At time \(t = 0\) seconds, a particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle \(60°\) above the horizontal from a point \(O\). In the subsequent motion \(P\) moves freely under gravity. The direction of motion of \(P\) when \(t = 5\) is perpendicular to its direction of motion when \(t = 15\). Find the value of \(u\). [5]

A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. Its initial speed is \(u\) ms\(^{-1}\) and its angle of projection is \(\sin^{-1}(\frac{3}{5})\) above the horizontal. At time 8 s after projection, \(P\) is at the point \(A\). At time 32 s after projection, \(P\) is at the point \(B\). The direction of motion of \(P\) at \(B\) is perpendicular to its direction of motion at \(A\). Find the value of \(u\). [7]

A particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane. The particle moves freely under gravity.

1. Write down the horizontal and vertical components of the velocity of the particle at time \(T\) after projection. [2] At time \(T\) after projection, the direction of motion of the particle is perpendicular to the direction of projection.
2. Express \(T\) in terms of \(u\), \(g\) and \(\alpha\). [2]
3. Deduce that \(T > \frac{u}{g}\). [1]

A small boat \(S\), drifting in the sea, 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 \((4\mathbf{i} - 6\mathbf{j})\) km relative to a fixed origin \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. At 0945, \(S\) is at the point with position vector \((7\mathbf{i} - 7.5\mathbf{j})\) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf{s}\) km.

1. Calculate the bearing on which \(S\) is drifting. [4]
2. Find an expression for \(\mathbf{s}\) in terms of \(t\). [3]

At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity \((p\mathbf{i} + q\mathbf{j})\) km h\(^{-1}\). Given that \(M\) intercepts \(S\) at 1015,

1. calculate the value of \(p\) and the value of \(q\). [6]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).] Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \((2\mathbf{i} - 3\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) moves with velocity \((3\mathbf{i} + 4\mathbf{j})\) km h\(^{-1}\).

1. Find, to the nearest degree, the bearing on which \(Q\) is moving. [2]

At 2 pm, ship \(P\) is at the point with position vector \((\mathbf{i} + \mathbf{j})\) km and ship \(Q\) is at the point with position vector \((-2\mathbf{j})\) km. At time \(t\) hours after 2 pm, the position vector of \(P\) is \(\mathbf{p}\) km and the position vector of \(Q\) is \(\mathbf{q}\) km.

1. Write down expressions, in terms of \(t\), for
   1. \(\mathbf{p}\),
   2. \(\mathbf{q}\),
   3. \(\overrightarrow{PQ}\). [5]
2. Find the time when
   1. \(Q\) is due north of \(P\),
   2. \(Q\) is north-west of \(P\). [4]