Position vector at time t (constant velocity)

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

A ship \(S\) is moving with constant velocity \(( - 2.5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time 1200, the position vector of \(S\) relative to a fixed origin \(O\) is \(( 16 \mathbf { i } + 5 \mathbf { j } )\) km. Find

1. the speed of \(S\),
2. the bearing on which \(S\) is moving. The ship is heading directly towards a submerged rock \(R\). A radar tracking station calculates that, if \(S\) continues on the same course with the same speed, it will hit \(R\) at the time 1500.
3. Find the position vector of \(R\). The tracking station warns the ship's captain of the situation. The captain maintains \(S\) on its course with the same speed until the time is 1400 . He then changes course so that \(S\) moves due north at a constant speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming that \(S\) continues to move with this new constant velocity, find
4. an expression for the position vector of the ship \(t\) hours after 1400,
5. the time when \(S\) will be due east of \(R\),
6. the distance of \(S\) from \(R\) at the time 1600.

6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
3. Find the distance of \(S\) from \(B\) when \(t = 3\)
4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).

1. A particle \(P\) is moving with constant velocity. The position vector of \(P\) at time \(t\) seconds \(( t \geqslant 0 )\) is \(\mathbf { r }\) metres, relative to a fixed origin \(O\), and is given by

$$\mathbf { r } = ( 2 t - 3 ) \mathbf { i } + ( 4 - 5 t ) \mathbf { j }$$

1. Find the initial position vector of \(P\). The particle \(P\) passes through the point with position vector \(( 3.4 \mathbf { i } - 12 \mathbf { j } )\) m at time \(T\) seconds.
2. Find the value of \(T\).
3. Find the speed of \(P\).

8. \begin{figure}[h] \end{figure} A square floor space \(A B C D\), with centre \(O\), is modelled as a flat horizontal surface measuring 50 m by 50 m , as shown in Figure 5 .
The horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the direction of \(\overrightarrow { A B }\) and \(\overrightarrow { A D }\) respectively.
All position vectors are given relative to \(O\).
A small robot \(R\) is programmed to travel across the floor at a constant velocity.

• At time \(t = 0 , R\) is at the point with position vector ( \(- 2 \mathbf { i } + \mathbf { j }\) ) m
• At time \(t = 11 \mathrm {~s} , R\) is at the point with position vector \(( 9 \mathbf { i } + 23 \mathbf { j } ) \mathrm { m }\)
• At time \(t\) seconds, the position vector of \(R\) is \(\mathbf { r }\) metres
  1. Find, in terms of \(t\), i and j, an expression for \(\mathbf { r }\)

A second robot \(S\) is at the point \(C\).

$$\overrightarrow { S R } = [ ( 2 t - 27 ) \mathbf { i } + ( 3 t - 24 ) \mathbf { j } ] \mathbf { m }$$

Find the time when the distance between \(R\) and \(S\) is a minimum.

1. \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

Two speedboats, \(A\) and \(B\), are each moving with constant velocity.

• the velocity of \(A\) is \(40 \mathrm { kmh } ^ { - 1 }\) due east
• the velocity of \(B\) is \(20 \mathrm { kmh } ^ { - 1 }\) on a bearing of angle \(\alpha \left( 0 ^ { \circ } < \alpha < 90 ^ { \circ } \right)\), where \(\tan \alpha = \frac { 4 } { 3 }\) The boats are modelled as particles.
  1. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) in \(\mathrm { km } \mathrm { h } ^ { - 1 }\)

At noon

• the position vector of \(A\) is \(20 \mathbf { j } \mathrm {~km}\)
• the position vector of \(B\) is \(( 10 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\)

At time \(t\) hours after noon

• the position vector of \(A\) is \(\mathbf { r k m }\), where \(\mathbf { r } = 20 \mathbf { j } + 40 t \mathbf { i }\)
• the position vector of \(B\) is \(\mathbf { s }\) km
• Find an expression for \(\mathbf { s }\) in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
• Show that at time \(t\) hours after noon,

$$\overrightarrow { A B } = [ ( 10 - 24 t ) \mathbf { i } + ( 12 t - 15 ) \mathbf { j } ] \mathrm { km }$$

Show that the boats will never collide.

Find the distance between the boats when the bearing of \(B\) from \(A\) is \(225 ^ { \circ }\)

5.
[In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal vectors due east and due north respectively and position vectors are given relative to a fixed origin.]

1. The position vector, \(\mathbf { r }\) metres, of a particle \(P\) at time \(t\) seconds, relative to a fixed origin \(O\), is given by

$$\mathbf { r } = ( t - 3 ) \mathbf { i } + ( 1 - 2 t ) \mathbf { j }$$

1. Find, to the nearest degree, the size of the angle between \(\mathbf { r }\) and the vector \(\mathbf { j }\), when \(t = 2\)
2. Find the values of \(t\) for which the distance of \(P\) from \(O\) is 2.5 m .

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.] At 7 am a ship leaves a port and moves with constant velocity. The position vector of the port is \(( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At 7.36 am the ship is at the point with position vector \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }\).

1. Show that the velocity of the ship is \(( 10 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
2. Find the position vector of the ship \(t\) hours after leaving port. At 8.48 am a passenger on the ship notices that a lighthouse is due east of the ship. At 9 am the same passenger notices that the lighthouse is now north east of the ship.
3. Find the position vector of the lighthouse.
4. Find the position vector of the ship when it is due south of the lighthouse.
   
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6. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.] A particle \(P\) is moving with constant velocity \(( - 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the speed of \(P\),
2. the direction of motion of \(P\), giving your answer as a bearing. At time \(t = 0 , P\) is at the point \(A\) with position vector ( \(7 \mathbf { i } - 10 \mathbf { j }\) ) m relative to a fixed origin \(O\). When \(t = 3 \mathrm {~s}\), the velocity of \(P\) changes and it moves with velocity \(( u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(u\) and \(v\) are constants. After a further 4 s , it passes through \(O\) and continues to move with velocity ( \(u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
3. Find the values of \(u\) and \(v\).
4. Find the total time taken for \(P\) to move from \(A\) to a position which is due south of A.

8 In this question, positions are given relative to a fixed origin, O. The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).

1. Find the distance AB .
2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) }$$
3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
4. What can you say about the result of the race?
5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.

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

A stone slides horizontally across ice.
Initially the stone is at the point \(A ( - 24 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }\) relative to a fixed point \(O\).
After 4 seconds the stone is at the point \(B ( 12 \mathbf { i } + 5 \mathbf { j } )\) m relative to the fixed point \(O\).
The motion of the stone is modelled as that of a particle moving in a straight line at constant speed. Using the model,

1. prove that the stone passes through \(O\),
2. calculate the speed of the stone.

4. The position of an aeroplane flying in a straight horizontal line at constant speed is plotted on a radar screen. At 2 p.m. the position vector of the aeroplane is \(( 80 \mathbf { i } + 5 \mathbf { j } )\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors directed east and north respectively relative to a fixed origin, \(O\), on the screen. Ten minutes later the position of the aeroplane on the screen is \(( 32 \mathbf { i } + 19 \mathbf { j } )\). Each unit on the screen represents 1 km .

1. Find the position vector of the aeroplane at 2:30 p.m.
2. Find the speed of the aeroplane in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
3. Find, correct to the nearest degree, the bearing on which the aeroplane is flying.

5 In this question, positions are given relative to a fixed origin, O . The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).

1. Find the distance AB .
2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) } .$$
3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
4. What can you say about the result of the race?
5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.

2 In this question, positions are given relative to a fixed origin, O. The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).

1. Find the distance AB .
2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) }$$
3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
4. What can you say about the result of the race?
5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.

A particle \(P\) is moving with constant velocity \((-3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 6\) s \(P\) is at the point with position vector \((-4\mathbf{i} - 7\mathbf{j})\) m. Find the distance of \(P\) from the origin at time \(t = 2\) s. [5]

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors.

1. Find the distance \(XY\). [2 marks]

A particle \(P\) of mass \(2\) kg moves from \(X\) to \(Y\) in \(4\) seconds, in a straight line at a constant speed.

1. Show that the velocity vector of \(P\) is \((2\mathbf{i} + 1.5\mathbf{j}) \text{ ms}^{-1}\). [3 marks]

The particle continues beyond \(Y\) with the same constant velocity.

1. Write down an expression for the position vector of \(P\) \(t\) seconds after leaving \(X\). [2 marks]
2. Find the value of \(t\) when \(P\) is at the point with position vector \((16\mathbf{i} + 4\mathbf{j})\) m. [2 marks]

When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass \(4\) kg, which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.

1. Find the velocity vector of the combined particle after the collision. [5 marks]