1.10h Vectors in kinematics: uniform acceleration in vector form

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.

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

A particle \(P\) moves with constant acceleration \(( - \lambda \mathbf { i } + 2 \lambda \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), where \(\lambda\) is a positive constant. At time \(t = 0\), the velocity of \(P\) is \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the velocity of \(P\) when \(t = 5 \mathrm {~s}\), giving your answer in terms of \(\mathbf { i } , \mathbf { j }\) and \(\lambda\). The speed of \(P\) when \(t = 5 \mathrm {~s}\) is \(13 \mathrm {~ms} ^ { - 1 }\)
2. Show that $$25 \lambda ^ { 2 } - 42 \lambda - 16 = 0$$
3. Find the direction of motion of \(P\) when \(t = 4 \mathrm {~s}\), giving your answer as a bearing to the nearest degree.

1. \hspace{0pt} [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 \(O\).]

At 12:00, a ship \(P\) sets sail from a harbour with position vector \(( 15 \mathbf { i } + 36 \mathbf { j } ) \mathrm { km }\). At 12:30, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). Given that \(P\) moves with constant velocity,

1. show that the velocity of \(P\) is \(( 10 \mathbf { i } - 4 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) At time \(t\) hours after 12:00, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).
2. Find an expression for \(\mathbf { p }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A second ship \(Q\) is also travelling at a constant velocity.
   At time \(t\) hours after 12:00, the position vector of \(Q\) is given by \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = ( 42 - 8 t ) \mathbf { i } + ( 9 + 14 t ) \mathbf { j }$$ Ships \(P\) and \(Q\) are modelled as particles.
   If both ships maintained their course,
3. 1. verify that they would collide at 13:30
   2. find the position vector of the point at which the collision would occur. At 12:30 \(Q\) changes speed and direction to avoid the collision.
      Ship \(Q\) now travels due north with a constant speed of \(15 \mathrm { kmh } ^ { - 1 }\) Ship \(P\) maintains the same constant velocity throughout.
4. Find the exact distance between \(P\) and \(Q\) at 14:30

4. A particle \(P\) moves in a straight line with constant velocity. Initially \(P\) is at the point \(A\) with position vector \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\), and 2 s later it is at the point \(B\) with position vector \(( 6 \mathbf { i } + \mathbf { j } ) \mathrm { m }\).

1. Find the velocity of \(P\).
2. Find, in degrees to one decimal place, the size of the angle between the direction of motion of \(P\) and the vector \(\mathbf { i }\).
   (2 marks)
   Three seconds after it passes \(B\) the particle \(P\) reaches the point \(C\).
3. Find, in m to one decimal place, the distance \(O C\).

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.

1. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0 , P\) has speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 3 \mathrm {~s} , P\) has velocity \(( - 6 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the value of \(u\).
(5)

6 The velocity of a model boat, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \binom { - 5 } { 10 } + t \binom { 6 } { - 8 }$$ where \(t\) is the time in seconds and the vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are east and north respectively.

1. Show that when \(t = 2.5\) the boat is travelling south-east (i.e. on a bearing of \(135 ^ { \circ }\) ). Calculate its speed at this time. The boat is at a point O when \(t = 0\).
2. Calculate the bearing of the boat from O when \(t = 2.5\).

2 In this question, the unit vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north.
Distance is measured in metres and time, \(t\), in seconds.
A radio-controlled toy car moves on a flat horizontal surface. A child is standing at the origin and controlling the car.
When \(t = 0\), the displacement of the car from the origin is \(\binom { 0 } { - 2 } \mathrm {~m}\), and the car has velocity \(\binom { 2 } { 0 } \mathrm {~ms} ^ { - 1 }\). The acceleration of the car is constant and is \(\binom { - 1 } { 1 } \mathrm {~ms} ^ { - 2 }\).

1. Find the velocity of the car at time \(t\) and its speed when \(t = 8\).
2. Find the distance of the car from the child when \(t = 8\).

5 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } ,$$ referred to an origin \(\mathbf { O }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes \(\mathrm { O } x\) and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both \(\mathrm { O } x\) and \(\mathrm { O } y\). Section B (36 marks)

4 Fig. 4 shows the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) in the directions of the cartesian axes \(\mathrm { O } x\) and \(\mathrm { O } y\), respectively. O is the origin of the axes and of position vectors. \begin{figure}[h] \end{figure} The position vector of a particle is given by \(\mathbf { r } = 3 t \mathbf { i } + \left( 18 t ^ { 2 } - 1 \right) \mathbf { j }\) for \(t \geqslant 0\), where \(t\) is time.

1. Show that the path of the particle cuts the \(x\)-axis just once.
2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the j direction.
3. Find the cartesian equation of the path of the particle, simplifying your answer.

6 A rock of mass 8 kg is acted on by just the two forces \(- 80 \mathbf { k } \mathrm {~N}\) and \(( - \mathbf { i } + 16 \mathbf { j } + 72 \mathbf { k } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upward.

1. Show that the acceleration of the rock is \(\left( - \frac { 1 } { 8 } \mathbf { i } + 2 \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 2 }\). The rock passes through the origin of position vectors, O , with velocity \(( \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and 4 seconds later passes through the point A .
2. Find the position vector of A .
3. Find the distance OA .
4. Find the angle that OA makes with the horizontal. Section B (36 marks)

3 An object of mass 5 kg has a constant acceleration of \(\binom { - 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(0 \leqslant t \leqslant 4\), where \(t\) is the time in seconds.

1. Calculate the force acting on the object. When \(t = 0\), the object has position vector \(\binom { - 2 } { 3 } \mathrm {~m}\) and velocity \(\binom { 4 } { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the position vector of the object when \(t = 4\).

5 The position vector of a toy boat of mass 1.5 kg is modelled as \(\mathbf { r } = ( 2 + t ) \mathbf { i } + \left( 3 t - t ^ { 2 } \right) \mathbf { j }\) where lengths are in metres, \(t\) is the time in seconds, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal, perpendicular unit vectors and the origin is O .

1. Find the velocity of the boat when \(t = 4\).
2. Find the acceleration of the boat and the horizontal force acting on the boat.
3. Find the cartesian equation of the path of the boat referred to \(x\) - and \(y\)-axes in the directions of \(\mathbf { i }\) and \(\mathbf { j }\), respectively, with origin O . You are not required to simplify your answer. Section B (36 marks)

3 The map of a large area of open land is marked in 1 km squares and a point near the middle of the area is defined to be the origin. The vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north. At time \(t\) hours the position vectors of two hikers, Ashok and Kumar, are given by: $$\begin{array} { l l } \text { Ashok } & \mathbf { r } _ { \mathrm { A } } = \binom { - 2 } { 0 } + \binom { 8 } { 1 } t , \\ \text { Kumar } & \mathbf { r } _ { \mathrm { K } } = \binom { 7 t } { 10 - 4 t } . \end{array}$$

1. Prove that the two hikers meet and give the coordinates of the point where this happens.
2. Compare the speeds of the two hikers.

1. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - 8 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - t ^ { 2 } + 2 t \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 4\) At time \(T\) seconds, \(T \geqslant 0 , P\) is moving in the direction of ( \(2 \mathbf { i } + \mathbf { j }\) )
2. Find the value of \(T\)

1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A particle \(P\) is moving on a smooth horizontal plane.
At time \(t\) seconds \(( t \geqslant 0 )\), the position vector of \(P\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(P\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where $$\mathbf { v } = \left( 4 t ^ { 2 } - 5 t \right) \mathbf { i } + ( - 10 t - 12 ) \mathbf { j }$$ When \(t = 0 , \mathbf { r } = 2 \mathbf { i } + 6 \mathbf { j }\)

1. Find \(\mathbf { r }\) when \(t = 2\) When \(t = T\) particle \(P\) is moving in the direction of the vector \(\mathbf { i } - 2 \mathbf { j }\)
2. Find the value of \(T\)
3. Find the exact magnitude of the acceleration of \(P\) when \(t = 2.5\)

3. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ When \(t = 0\) the particle \(P\) is at the origin \(O\). At time \(T\) seconds, \(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\), where \(\lambda\) is a constant. Find

1. the value of \(T\),
2. the acceleration of \(P\) as it passes through the point \(A\),
3. the distance \(O A\).
   

1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]

\begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) is projected from the point \(A\) which has position vector 10j metres with respect to a fixed origin \(O\) at ground level. The ground is horizontal. The velocity of projection of \(P\) is \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 3. The particle moves freely under gravity and reaches the ground after \(T\) seconds.

1. For \(0 \leqslant t \leqslant T\), show that, with respect to \(O\), the position vector, \(\mathbf { r }\) metres, of \(P\) at time \(t\) seconds is given by $$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$
2. Find the value of \(T\).
3. Find the velocity of \(P\) at time \(t\) seconds \(( 0 \leqslant t \leqslant T )\). When \(P\) is at the point \(B\), the direction of motion of \(P\) is \(45 ^ { \circ }\) below the horizontal.
4. Find the time taken for \(P\) to move from \(A\) to \(B\).
5. Find the speed of \(P\) as it passes through \(B\).

4. At time \(t\) seconds the velocity of a particle \(P\) is \([ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(P\) is \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\), relative to a fixed origin \(O\).

1. Find the value of \(t\) when the velocity of \(P\) is parallel to the vector \(\mathbf { j }\).
2. Find an expression for the position vector of \(P\) at time \(t\) seconds. A second particle \(Q\) moves with constant velocity \(( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\). The particles \(P\) and \(Q\) collide at the point with position vector ( \(d \mathbf { i } + 14 \mathbf { j }\) ) m.
3. Find
   1. the value of \(c\),
   2. the value of \(d\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A particle \(P\) moves in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 1 \right) \mathbf { i } + \left( 4 t - t ^ { 2 } \right) \mathbf { j }$$

1. Find the magnitude of the acceleration of \(P\) when \(t = 1\) Given that, when \(t = 0\), the position vector of \(P\) is i metres,
2. find the position vector of \(P\) when \(t = 3\)

2. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) is moving on a horizontal plane with acceleration \(\left[ \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + ( 6 t - 5 ) \mathbf { j } \right] \mathrm { m } \mathrm { s } ^ { - 2 }\). When \(t = 3\) the velocity of \(P\) is \(( 11 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the speed of \(P\) when it is moving parallel to the vector \(\mathbf { i }\).

3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find

1. the value of \(c\) ,
2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D墐
   (b)the acceleration of \(P\) when \(t = 1.5\) .

1. An aeroplane leaves a runway and moves with a constant speed of \(V \mathrm {~km} / \mathrm { h }\) due north along a straight path inclined at an angle \(\arctan \left( \frac { 3 } { 4 } \right)\) to the horizontal.

A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway. The light aircraft is travelling with a constant speed of \(2 V \mathrm {~km} / \mathrm { h }\).
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance \(d \mathrm {~km}\) behind the aeroplane. Both aircraft continue to move with the same trajectories due north.

1. Show that the distance, \(D \mathrm {~km}\), between the two aircraft \(t\) hours after the aeroplane leaves the runway satisfies $$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$ Given that the distance between the two aircraft is never less than 2 km ,
2. find the range of possible values for \(d\).

3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find

1. the value of \(c\) ,
2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D啨
   (b)the acceleration of \(P\) when \(t = 1.5\) .