Position vector from velocity integration

5. A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find, to the nearest degree, the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 0\). At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
2. an expression for \(\mathbf { v }\) in terms of \(t\), in the form \(a \mathbf { i } + b \mathbf { j }\),
3. the speed of \(P\) when \(t = 3\),
4. the time when \(P\) is moving parallel to \(\mathbf { i }\).

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 \(O\).
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5. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = \left( 5 t ^ { 2 } - 12 t + 15 \right) \mathbf { i } + \left( t ^ { 2 } + 8 t - 10 \right) \mathbf { j }$$ When \(t = 0 , P\) is at the origin \(O\).
At time \(T\) seconds, \(P\) is moving in the direction of \(( \mathbf { i } + \mathbf { j } )\).

1. Find the value of \(T\). When \(t = 3 , P\) is at the point \(A\).
2. Find the magnitude of the acceleration of \(P\) as it passes through \(A\).
3. Find the position vector of \(A\).

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. At time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 9 t + 6 \right) \mathbf { i } + \left( t ^ { 2 } + t - 6 \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 3\) When \(t = 0 , P\) is at the fixed point \(O\).
   The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

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 \(t = 2.5\)

4. At time \(t\) seconds \(( 0 \leqslant t < 5 )\), a particle \(P\) has velocity \(\mathbf { v m s } ^ { - 1 }\), where $$\mathbf { v } = ( \sqrt { 5 - t } ) \mathbf { i } + \left( t ^ { 2 } + 2 t - 3 \right) \mathbf { j }$$ When \(t = \lambda\), particle \(P\) is moving in a direction parallel to the vector \(\mathbf { i }\).

1. Find the acceleration of \(P\) when \(t = \lambda\) The position vector of \(P\) is measured relative to the fixed point \(O\) When \(t = 1\), the position vector of \(P\) is \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { m }\). Given that \(1 \leqslant T < 5\)
2. find, in terms of \(T\), the position vector of \(P\) when \(t = T\)

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 \(t = 3\)

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. \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. 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. 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 \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

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 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.

8 A Jet Ski is at the origin and is travelling due north at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it begins to accelerate uniformly. After accelerating for 40 seconds, it is travelling due east at \(4 \mathrm {~ms} ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Show that the acceleration of the Jet Ski is \(( 0.1 \mathbf { i } - 0.125 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the position vector of the Jet Ski at the end of the 40 second period.
3. The Jet Ski is travelling southeast \(t\) seconds after it leaves the origin.
   1. Find \(t\).
   2. Find the velocity of the Jet Ski at this time.

5 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A helicopter moves horizontally with a constant acceleration of \(( - 0.4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the helicopter is at the origin and has velocity \(20 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Write down an expression for the velocity of the helicopter at time \(t\) seconds.
2. Find the time when the helicopter is travelling due north.
3. Find an expression for the position vector of the helicopter at time \(t\) seconds.
4. When \(t = 100\) :
   1. show that the helicopter is due north of the origin;
   2. find the speed of the helicopter.

1 The velocity of a particle at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 4 + 3 t ^ { 2 } \right) \mathbf { i } + ( 12 - 8 t ) \mathbf { j }$$

1. When \(t = 0\), the particle is at the point with position vector \(( 5 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
2. Find the acceleration of the particle at time \(t\).
3. The particle has mass 2 kg . Find the magnitude of the force acting on the particle when \(t = 1\).

3. A particle \(P\) moves in a horizontal plane such that, at time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - t ^ { \frac { 1 } { 2 } } \mathbf { j }\). When \(t = 0 , P\) is at the point with position vector \(- 10 \mathbf { i } + \mathbf { j }\) relative to a fixed origin \(O\).

1. Find the position vector \(\mathbf { r }\) of \(P\) at time \(t\) seconds.
2. Find the distance \(O P\) when \(t = 4\).

3 A particle Q of mass \(m\) moves in a horizontal plane under the action of a single force \(\mathbf { F }\). At time \(t , \mathrm { Q }\) has velocity \(\binom { 2 } { 3 t - 2 }\).

1. Find an expression for \(\mathbf { F }\) in terms of \(m\). At time \(t\), the displacement of Q is given by \(\mathbf { r } = \binom { x } { y }\). When \(t = 1 , \mathrm { Q }\) is at the point with position vector \(\binom { 4 } { - 4 }\).
2. Find the equation of the path of Q , giving your answer in the form \(y = a x ^ { 2 } + b x + c\), where \(a\), \(b\) and \(c\) are constants to be determined.
3. What can you deduce about the path of Q from the value of the constant \(c\) you found in part (b)?

4 A particle moves so that at time \(t\) seconds its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = \left( 4 t ^ { 3 } - 12 t + 3 \right) \mathbf { i } + 5 \mathbf { j } + 8 t \mathbf { k }$$

1. When \(t = 0\), the position vector of the particle is \(( - 5 \mathbf { i } + 6 \mathbf { k } )\) metres. Find the position vector of the particle at time \(t\).
2. Find the acceleration of the particle at time \(t\).
3. Find the magnitude of the acceleration of the particle at time \(t\). Do not simplify your answer.
4. Hence find the time at which the magnitude of the acceleration is a minimum.
5. The particle is moving under the action of a single variable force \(\mathbf { F }\) newtons. The mass of the particle is 7 kg . Find the minimum magnitude of \(\mathbf { F }\).

15 At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally.
The velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the parachutist at time \(t\) seconds is given by: $$\mathbf { v } = \left( 40 \mathrm { e } ^ { - 0.2 t } \right) \mathbf { i } + 50 \left( \mathrm { e } ^ { - 0.2 t } - 1 \right) \mathbf { j }$$ The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.
Assume that the parachutist is at the origin when \(t = 0\)
Model the parachutist as a particle. 15

1. Find an expression for the position vector of the parachutist at time \(t\).
   [0pt] [4 marks] 15
2. The parachutist opens her parachute when she has travelled 100 metres horizontally.
   Find the vertical displacement of the parachutist from the origin when she opens her parachute.
   [0pt] [4 marks]
   15
3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model.

6 In this question you must show detailed reasoning. In this question, positions are given relative to a fixed origin, O . The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively in a horizontal plane. Distances are measured in centimetres and the time, \(t\), is measured in seconds, where \(0 \leqslant t \leqslant 5\). A small radio-controlled toy car C moves on a horizontal surface which contains O.
The acceleration of C is given by \(2 \mathbf { i } + t \mathbf { j } \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
When \(t = 4\), the displacement of C from O is \(16 \mathbf { i } + \frac { 32 } { 3 } \mathbf { j } \mathrm {~cm}\), and the velocity of C is \(8 \mathbf { i } \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
Determine a cartesian equation for the path of C for \(0 \leqslant t \leqslant 5\). You are not required to simplify your answer.