Force from vector acceleration

3 A small block \(B\) of mass 0.2 kg is placed at a fixed point \(O\) on a smooth horizontal surface. A horizontal force of magnitude 0.42 N is applied to \(B\). At time \(t \mathrm {~s}\) after the force is first applied, the velocity of \(B\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(v\) when \(t = 1\). For \(t > 1\) an additional force, of magnitude \(0.32 t \mathrm {~N}\) and directed towards \(O\), is applied to \(B\). The force of magnitude 0.42 N continues to act as before.
2. Find the value of \(v\) when \(t = 2\). For \(t > 2\) a third force, of magnitude \(0.06 t ^ { 2 } \mathrm {~N}\) and directed away from \(O\), is applied to \(B\). The other two forces continue to act as before.
3. Show that the velocity of \(B\) is the same when \(t = 2\) and when \(t = 3\).

2. A particle \(P\) of mass 1.5 kg is moving under the action of a constant force ( \(3 \mathbf { i } - 7.5 \mathbf { j }\) ) N. Initially \(P\) has velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the magnitude of the acceleration of \(P\),
2. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\), when \(P\) has been moving for 4 seconds.

3. A particle \(P\) of mass 0.25 kg is moving on a smooth horizontal surface under the action of a single force, \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 )\), the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = ( 6 \sin 3 t ) \mathbf { i } + ( 1 + 2 \cos t ) \mathbf { j }$$

1. Find \(\mathbf { F }\) in terms of \(t\). At time \(t = 0\), the position vector of \(P\) relative to a fixed point \(O\) is \(( 4 \mathbf { i } - \sqrt { 3 } \mathbf { j } ) \mathrm { m }\).
2. Find the position vector of \(P\) relative to \(O\) when \(P\) is first moving parallel to the vector \(\mathbf { i }\).

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

A particle \(Q\) of mass 1.5 kg is moving on a smooth horizontal plane under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds ( \(t \geqslant 0\) ), the position vector of \(Q\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(Q\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\)
It is given that $$\mathbf { v } = \left( 3 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t ^ { 3 } + k t \right) \mathbf { j }$$ where \(k\) is a constant.
Given that when \(t = 2\) particle \(Q\) is moving in the direction of the vector \(\mathbf { i } + \mathbf { j }\)

1. show that \(k = 4\)
2. find the magnitude of \(\mathbf { F }\) when \(t = 2\) Given that \(\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }\) when \(t = 0\)
3. find \(\mathbf { r }\) when \(t = 2\)

1 The position vector, \(\mathbf { r }\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.

1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
2. Find \(\mathbf { F }\).

5 The acceleration of a particle of mass 4 kg is given by \(\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors and \(t\) is the time in seconds.

1. Find the acceleration of the particle when \(t = 0\) and also when \(t = 3\).
2. Calculate the force acting on the particle when \(t = 3\). The particle has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when \(t = 1\).
3. Find an expression for the velocity of the particle at time \(t\).

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. A particle \(P\) of mass 0.3 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 ) , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + \left( 3 t ^ { 2 } - 8 t + 4 \right) \mathbf { j }$$

1. Find \(\mathbf { F }\) when \(t = 4\) At the instants when \(P\) is at the points \(A\) and \(B\), particle \(P\) is moving parallel to the vector i.
2. Find the distance \(A B\).
   

2. A particle \(P\) of mass 1.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = \left( 5 t ^ { 2 } - t ^ { 3 } \right) \mathbf { i } + \left( 2 t ^ { 3 } - 8 t \right) \mathbf { j }$$

1. Find \(\mathbf { F }\) when \(t = 2\) At time \(t = 0 , P\) is at the origin \(O\).
2. Find the position vector of \(P\) relative to \(O\) at the instant when \(P\) is moving in the direction of the vector \(\mathbf { j }\)

4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j } .$$ When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\).

1. Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
2. Calculate the distance \(O S\).

2. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } + ( 1 - 4 t ) \mathbf { j }$$ Find

1. the acceleration of \(P\) at time \(t\) seconds,
2. the magnitude of \(\mathbf { F }\) when \(t = 2\).

3 A particle \(P\) of mass 0.25 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = ( 2 - 4 t ) \mathbf { i } + \left( t ^ { 2 } + 2 t \right) \mathbf { j }$$ When \(t = 0 , P\) is at the point with position vector ( \(2 \mathbf { i } - 4 \mathbf { j }\) ) m with respect to a fixed origin \(O\). When \(t = 3 , P\) is at the point \(A\). Find

1. the momentum of \(P\) when \(t = 3\),
2. the magnitude of \(\mathbf { F }\) when \(t = 3\),
3. the position vector of \(A\).

1. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 4 t - 3 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 8 t - 40 \right) \mathbf { j }$$

1. Find
   1. the magnitude of \(\mathbf { F }\) when \(t = 3\)
   2. the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(- \mathbf { i } - \mathbf { j }\). When \(t = 1 , P\) is at the point \(A\). When \(t = 2 , P\) is at the point \(B\).
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the vector \(\overrightarrow { A B }\).

12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf { F N }\). At time \(t\) seconds, where \(t \geqslant 0\), the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(P\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = ( 1 - 2 t ) \mathbf { i } + \left( 2 t ^ { 2 } + t - 13 \right) \mathbf { j }\).

1. Show that \(P\) is never stationary.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the acceleration of \(P\) at time \(t\). The mass of \(P\) is 0.5 kg .
3. Determine the magnitude of \(\mathbf { F }\) when \(P\) is moving in the direction of the vector \(- 2 \mathbf { i } + \mathbf { j }\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When \(t = 1 , P\) is at the point with position vector \(\frac { 1 } { 6 } \mathbf { j }\).
4. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\).

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.

1 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 \(\left( \frac { 1 } { 8 } \mathbf { i } + 2 \mathbf { j } \quad \mathbf { k } \right) \mathrm { ms } ^ { - 2 }\). 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.

3 The position vector, \(r\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.

1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
2. Find \(\mathbf { F }\).

5 A particle moves such that at time \(t\) seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$

1. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 0\).
2. When \(t = 0\), the velocity of the particle is \(( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find an expression for the velocity of the particle at time \(t\).

4 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector, \(\mathbf { r }\) metres, of the particle is given by $$\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } + 4 \right) \mathbf { i } + \left( 4 t + t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. The mass of the particle is 3 kg .
   1. Find an expression for \(\mathbf { F }\) at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due north.

2 A particle moves in a horizontal plane. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 12 \cos \left( \frac { \pi } { 3 } t \right) \mathbf { i } - 9 t ^ { 2 } \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The particle, which has mass 4 kg , moves under the action of a single force, \(\mathbf { F }\) newtons.
   1. Find an expression for the force \(\mathbf { F }\) in terms of \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
3. When \(t = 3\), the particle is at the point with position vector \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).

3 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \mathrm { e } ^ { - 2 t } \mathbf { i } + \left( 6 t - 3 t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 5 kg .
   1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 0\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due west.
4. When \(t = 0\), the particle is at the point with position vector \(( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).

1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$

1. 1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
   2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}

2 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons.
The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \left( 8 t - t ^ { 4 } \right) \mathbf { i } + 6 \mathrm { e } ^ { - 3 t } \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 2 kg .
   1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 1\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due south.
4. When \(t = 0\), the particle is at the point with position vector \(( 3 \mathbf { i } - 5 \mathbf { j } )\) metres. Find an expression for the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
   [0pt] [4 marks]