3.02e Two-dimensional constant acceleration: with vectors

3. A particle \(P\) of mass 0.4 kg moves under the action of a single constant force \(\mathbf { F }\) newtons. The acceleration of \(P\) is \(( 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the angle between the acceleration and \(\mathbf { i }\),
2. the magnitude of \(\mathbf { F }\). At time \(t\) seconds the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when \(t = 0 , \mathbf { v } = 9 \mathbf { i } - 10 \mathbf { j }\), (c) find the velocity of \(P\) when \(t = 5\).

5. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\).

1. Show that the magnitude of the acceleration of \(P\) is \(2 \sqrt { 13 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the velocity of \(P\) at time \(t = 2\) seconds. Another particle \(Q\) moves with constant velocity \(\mathbf { v } = ( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
3. Find the distance moved by \(Q\) in 2 seconds.
4. Show that at time \(t = 3.5\) seconds both particles are moving in the same direction.

5. A t time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + ( 2 t - 4 ) \mathbf { j }$$ When \(t = 0 , P\) is at the fixed point \(O\).

1. Find the acceleration of \(P\) at the instant when \(t = 0\)
2. Find the exact speed of \(P\) at the instant when \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } + \mathbf { j } )\) for the second time.
3. Show that \(P\) never returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-14_2658_1938_107_123} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-15_149_140_2604_1818}

8. Two trains, \(A\) and \(B\), start together from rest, at time \(t = 0\), at a station and move along parallel straight horizontal tracks. Both trains come to rest at the next station after 180 s . Train \(A\) moves with constant acceleration \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 30 s , then moves at constant speed for 120 s and then moves with constant deceleration for the final 30 s . Train \(B\) moves with constant acceleration for 90 s and then moves with constant deceleration for the final 90 s .

1. Sketch, on the same axes, the speed-time graphs for the motion of the two trains between the two stations.
2. Find the acceleration of train \(B\) for the first half of its journey.
3. Find the times when the two trains are moving at the same speed.
4. Find the distance between the trains 96 s after they start. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-28_43_58_2457_1893}

2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] At time \(t = 0\), a bird \(A\) leaves its nest, that is located at the point with position vector \(( 20 \mathbf { i } - 17 \mathbf { j } ) \mathrm { m }\), and flies with constant velocity \(( - 6 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the same time a second bird \(B\) leaves its nest which is located at the point with position vector \(( - 8 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m }\) and flies with constant velocity ( \(p \mathbf { i } + 2 p \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), where \(p\) is a constant. At time \(t = 4 \mathrm {~s}\), bird \(B\) is south west of bird \(A\).

1. Find the direction of motion of \(A\), giving your answer as a bearing to the nearest degree.
2. Find the speed of \(B\).

6. Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles. At time \(t = 0\), Agatha is at the point with position vector \(( 11 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m }\) and Brionie is at the point with position vector \(( 7 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m }\). The position vectors are given relative to the door, \(O\), and \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. Agatha skates with constant velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) Brionie skates with constant velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the position vector of Agatha at time \(t\) seconds. At time \(t = 6\) seconds, Agatha passes through the point \(P\).
2. Show that Brionie also passes through \(P\) and find the value of \(t\) when this occurs. At time \(t\) seconds, Agatha is at the point \(A\) and Brionie is at the point \(B\).
3. Show that \(\overrightarrow { A B } = [ ( t - 4 ) \mathbf { i } + ( 5 - t ) \mathbf { j } ] \mathrm { m }\)
4. Find the distance between the two girls when they are closest together. \includegraphics[max width=\textwidth, alt={}, center]{ca445c1e-078c-4a57-94df-de90f30f8efd-13_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

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

A particle \(P\) of mass 2 kg moves under the action of two forces, \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) and \(( 2 q \mathbf { i } + p \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants. Given that the acceleration of \(P\) is \(( \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)

1. find the value of \(p\) and the value of \(q\).
2. Find the size of the angle between the direction of the acceleration and the vector \(\mathbf { j }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At \(t = T\) seconds, \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } - 13 \mathbf { j } )\).
3. Find the value of \(T\).

1. A particle \(P\) is moving with constant acceleration ( \(- 4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { ms } ^ { - 2 }\)

At time \(t = 0 , P\) has velocity \(( 14 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the speed of \(P\) at time \(t = 2\) seconds.
2. Find the size of the angle between the direction of \(\mathbf { i }\) and the direction of motion of \(P\) at time \(t = 2\) seconds. At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(2 \mathbf { i } - 3 \mathbf { j }\) )
3. Find the value of \(T\)

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

A boy \(B\) is running in a field with constant velocity ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0 , B\) is at the point with position vector 10j m . Find

1. the speed of \(B\),
2. the direction in which \(B\) is running, giving your answer as a bearing. At time \(t = 0\), a girl \(G\) is at the point with position vector \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). The girl is running with constant velocity \(\left( \frac { 5 } { 3 } \mathbf { i } + 2 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\) and meets \(B\) at the point \(P\).
3. Find
   1. the value of \(t\) when they meet,
   2. the position vector of \(P\).

1. A particle \(P\) moves from point \(A\) to point \(B\) with constant acceleration \(( c \mathbf { i } + d \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(c\) and \(d\) are positive constants. The velocity of \(P\) at \(A\) is \(( - 3 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(P\) at \(B\) is \(( 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The magnitude of the acceleration of \(P\) is \(2.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

Find the value of \(c\) and the value of \(d\).

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

A particle \(P\) moves with constant acceleration \(( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v m ~ s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = 10 \mathbf { i } + 4 \mathbf { j }\).

1. Find the direction of motion of \(P\) when \(t = 6\), giving your answer as a bearing to the nearest degree.
2. Find the value of \(t\) when \(P\) is moving north east.

6. A particle \(P\) is moving with constant acceleration. At time \(t = 1\) second, \(P\) has velocity \(( - \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At time \(t = 4\) seconds, \(P\) has velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Find the speed of \(P\) at time \(t = 3.5\) seconds.

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 A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\binom { - 1 } { 2 } \mathrm {~m}\) with velocity \(\binom { 2 } { - 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\); after 4 seconds the particle has velocity \(\binom { 12 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the acceleration of the particle.
2. Calculate the position of the particle at the end of the 4 seconds.
3. Calculate the force acting on the particle.

4 At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf { r } = \binom { 8 t } { 10 t ^ { 2 } - 2 t ^ { 3 } } ,$$ where \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are perpendicular unit vectors east and north respectively and distances are in metres.

1. When \(t = 1\), the particle is at P . Find the bearing of P from O .
2. Find the velocity of the particle at time \(t\) and show that it is never zero.
3. Determine the time(s), if any, when the acceleration of the particle is zero.

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 \(t = 8\).

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)

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)

4 A particle is initially at the origin, moving with velocity \(\mathbf { u }\). Its acceleration \(\mathbf { a }\) is constant. At time \(t\) its displacement from the origin is \(\mathbf { r } = \binom { x } { y }\), where \(\binom { x } { y } = \binom { 2 } { 6 } t - \binom { 0 } { 4 } t ^ { 2 }\).

1. Write down \(\mathbf { u }\) and \(\mathbf { a }\) as column vectors.
2. Find the speed of the particle when \(t = 2\).
3. Show that the equation of the path of the particle is \(y = 3 x - x ^ { 2 }\).

1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).

The particles are moving in the same direction along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is a second collision between \(P\) and \(Q\),
3. find the complete range of possible values of \(f\).

6. [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.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).

1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

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 \(d\).

6 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors east and north respectively. Position vectors are with respect to an origin O . Time \(t\) is in seconds. A skater has a constant acceleration of \(- 2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At \(t = 0\), his velocity is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his position vector is \(3 \mathbf { j } \mathrm {~m}\).

1. Find expressions in terms of \(t\) for the velocity and the position vector of the skater at time \(t\).
2. Calculate as a bearing the direction of motion of the skater when \(t = 2.5\).

8 In this question \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) denote unit vectors which are horizontal and vertically upwards respectively.
A particle of mass 5 kg , initially at rest at the point with position vector \(\binom { 2 } { 45 } \mathrm {~m}\), is acted on by gravity and also by two forces \(\binom { 15 } { - 8 } \mathrm {~N}\) and \(\binom { - 7 } { - 2 } \mathrm {~N}\).

1. Find the acceleration vector of the particle.
2. Find the position vector of the particle after 10 seconds.