1.10h Vectors in kinematics: uniform acceleration in vector form

8 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).

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, 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.

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.

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

A coastguard station \(O\) monitors the movements of a small boat.
At 10:00 the boat is at the point \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
At 12:45 the boat is at the point \(( - 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
The motion of the boat is modelled as that of a particle moving in a straight line at constant speed.

1. Calculate the bearing on which the boat is moving, giving your answer in degrees to one decimal place.
2. Calculate the speed of the boat, giving your answer in \(\mathrm { kmh } ^ { - 1 }\)

1. \hspace{0pt} [In this question position vectors are given relative to a fixed origin \(O\) ]

At time \(t\) seconds, where \(t \geqslant 0\), a particle, \(P\), moves so that its velocity \(\mathbf { v ~ m ~ s } ^ { - 1 }\) is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When \(t = 0\), the position vector of \(P\) is \(( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }\).

1. Find the acceleration of \(P\) when \(t = 4\)
2. Find the position vector of \(P\) when \(t = 4\)

1. A particle, \(P\), moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)

At time \(t = 0\), the particle is at the point \(A\) and is moving with velocity ( \(- \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(3 \mathbf { i } - 4 \mathbf { j }\) )

1. Find the value of \(T\). At time \(t = 4\) seconds, \(P\) is at the point \(B\).
2. Find the distance \(A B\).

1. \hspace{0pt} [In this question, position vectors are given relative to a fixed origin.] At time \(t\) seconds, where \(t > 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) where

$$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 6 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$

1. Find the speed of \(P\) at time \(t = 2\) seconds.
2. Find an expression, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\), for the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\) At time \(t = 4\) seconds, the position vector of \(P\) is ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
3. Find the position vector of \(P\) at time \(t = 1\) second.

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

A particle \(P\) of mass 4 kg is at rest at the point \(A\) on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }\), where \(\lambda\) and \(\mu\) are constants, are applied to \(P\) Given that \(P\) moves in the direction of the vector ( \(3 \mathbf { i } + \mathbf { j }\) )

1. show that $$\lambda - 3 \mu + 7 = 0$$ At time \(t = 4\) seconds, \(P\) passes through the point \(B\).
   Given that \(\lambda = 2\)
2. find the length of \(A B\).

1. A particle \(P\) moves with acceleration \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)

At time \(t = 0 , P\) is moving with velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the velocity of \(P\) at time \(t = 2\) seconds. At time \(t = 0 , P\) passes through the origin \(O\).
   At time \(t = T\) seconds, where \(T > 0\), the particle \(P\) passes through the point \(A\).
   The position vector of \(A\) is ( \(\lambda \mathbf { i } - 4.5 \mathbf { j }\) )m relative to \(O\), where \(\lambda\) is a constant.
2. Find the value of \(T\).
3. Hence find the value of \(\lambda\)

1. 1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration a \(\mathrm { ms } ^ { - 2 }\) is given by

$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when \(t = 0\), the velocity of \(P\) is \(36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) when \(t = 4\)
2. Find the value of \(t\) at the instant when \(P\) is moving in a direction perpendicular to i
   (ii) At time \(t\) seconds, where \(t \geqslant 0\), a particle \(Q\) moves so that its position vector \(\mathbf { r }\) metres, relative to a fixed origin \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of \(t\) at the instant when the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)

At time \(t = 0 , P\) is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) at time \(t = 2\) seconds. At time \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
2. Find the position vector of \(P\) relative to \(O\) at time \(t = 3\) seconds.

1. At time \(t\) seconds, a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where

$$\mathbf { v } = 3 t ^ { \frac { 1 } { 2 } } \mathbf { i } - 2 t \mathbf { j } \quad t > 0$$

1. Find the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\)
2. Find the value of \(t\) at the instant when \(P\) is moving in the direction of \(\mathbf { i } - \mathbf { j }\) At time \(t\) seconds, where \(t > 0\), the position vector of \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres. When \(t = 1 , \mathbf { r } = - \mathbf { j }\)
3. Find an expression for \(\mathbf { r }\) in terms of \(t\).
4. Find the exact distance of \(P\) from \(O\) at the instant when \(P\) is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. Determine whether the particle is ever stationary.

7 In this question the \(x\) - and \(y\)-directions are horizontal and vertically upwards respectively and the origin is on horizontal ground.
A ball is thrown from a point 5 m above the origin with an initial velocity \(\binom { 14 } { 7 } \mathrm {~ms} ^ { - 1 }\).

1. Find the position vector of the ball at time \(t \mathrm {~s}\) after it is thrown.
2. Find the distance between the origin and the point at which the ball lands on the ground.

9 In this question, the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
The velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of a particle at time \(t\) s is given by \(\mathbf { v } = k t ^ { 2 } \mathbf { i } + 6 t\), where \(k\) is a positive constant. The magnitude of the acceleration when \(t = 2\) is \(10 \mathrm {~ms} ^ { - 2 }\).

1. Calculate the value of \(k\). The particle is at the origin when \(t = 0\).
2. Determine an expression for the position vector of the particle at time \(t\).
3. Determine the time when the particle is directly north-east of the origin.

12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2 kg .

1. Write down its weight as a vector. A horizontal force of 3 N in the \(\mathbf { i }\) direction and a force \(\mathbf { F } = ( - 4 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\) act on the particle.
2. Determine the acceleration of the particle.
3. The initial velocity of the particle is \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\). Find the velocity of the particle after 4 s .
4. Find the extra force that must be applied to the particle for it to move at constant velocity.

12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively.
The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle is given by \(\mathbf { v } = 3 \mathbf { i } + \left( 6 t ^ { 2 } - 5 \right) \mathbf { j }\). The initial position of the particle is \(7 \mathbf { j } \mathrm {~m}\).

1. Find an expression for the position vector of the particle at time \(t \mathrm {~s}\).
2. Find the Cartesian equation of the path of the particle.

13 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the \(x\) - and \(y\)-directions respectively.
The velocity of a particle at time \(t \mathrm {~s}\) is given by \(\left( 3 t ^ { 2 } \mathbf { i } + 7 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\) the position of the particle with respect to the origin is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\).

1. Determine the distance of the particle from the origin when \(t = 2\).
2. Show that the cartesian equation of the path of the particle is \(x = \left( \frac { y - 2 } { 7 } \right) ^ { 3 } - 1\).
3. At time \(t = 2\), the magnitude of the resultant force acting on the particle is 48 N . Find the mass of the particle.

12 A model boat has velocity \(\mathbf { v } = ( ( 2 t - 2 ) \mathbf { i } + ( 2 t + 2 ) \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) for \(t \geq 0\), where \(t\) is the time in seconds. \(\mathbf { i }\) is the unit vector east and \(\mathbf { j }\) is the unit vector north.
When \(t = 3\), the position vector of the boat is \(( 3 \mathbf { i } + 14 \mathbf { j } ) \mathrm { m }\).

1. Show that the boat is never instantaneously at rest.
2. Determine any times at which the boat is moving directly northwards.
3. Determine any times at which the boat is north-east of the origin.

3 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).

1. Use the equation \(\mathbf { v . v } = \mathbf { u . u } + 2 \mathbf { a x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
2. 1. Show that the power initially generated by the force is - 8 mW .
   2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
3. 1. Find the time at which the power generated by the force is instantaneously zero.
   2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).

8 A golf ball is hit from a point on a horizontal surface, so that it has an initial velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The ball travels through the air and after 2.4 seconds hits a vertical wall at a height of 3 metres. The wall is at a horizontal distance of 38.4 metres from the point where the ball was hit. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-18_300_1000_566_520} Assume that the weight of the ball is the only force that acts on it as it travels through the air.

1. Find the horizontal component of the velocity of the ball.
2. \(\quad\) Find \(V\).
3. \(\quad\) Find \(\alpha\).

6 A ball is hit from horizontal ground with velocity \(( 10 \mathbf { i } + 24.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively.

1. State two assumptions that you should make about the ball in order to make predictions about its motion.
2. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_771_705_625}
   1. Show that the time of flight of the ball is 5 seconds.
   2. Find the range of the ball.
3. In fact the ball hits a vertical wall that is 20 metres from the initial position of the ball. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_403_1466_769} Find the height of the ball when it hits the wall.
4. If a heavier ball were projected in the same way, would your answers to part (b) of this question change? Explain why.

7 A particle moves on a smooth horizontal surface with acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). Initially the velocity of the particle is \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. Find the time when the particle is travelling in the \(\mathbf { i }\) direction.
3. Show that when \(t = 4\) the speed of the particle is \(20 \mathrm {~ms} ^ { - 1 }\).

6 The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 2 \mathbf { j } )\) metres and \(( 6 \mathbf { i } - 4 \mathbf { j } )\) metres respectively. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.

1. A particle moves from \(A\) to \(B\) with constant velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Calculate the time that the particle takes to move from \(A\) to \(B\).
2. The particle then moves from \(B\) to a point \(C\) with a constant acceleration of \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It takes 4 seconds to move from \(B\) to \(C\).
   1. Find the position vector of \(C\).
   2. Find the distance \(A C\).