Find force using F=ma

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. 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 \(\mathbf { j }\)

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 \(t = 2\).

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 \(\overrightarrow { A B }\).

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 \(\mathbf { F }\).

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.

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 \(t = 1\).

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]
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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 \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

1. Show that \(P\) is never stationary. [2]
2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]

The mass of \(P\) is 0.5 kg.

1. 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 3 significant figures. [5]

When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).

1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]