Given velocity function find force

7 A particle of mass 0.25 kg moves in a straight line on a smooth horizontal surface. A variable resisting force acts on the particle. At time \(t \mathrm {~s}\) the displacement of the particle from a point on the line is \(x \mathrm {~m}\), and its velocity is \(( 8 - 2 x ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It is given that \(x = 0\) when \(t = 0\).

1. Find the acceleration of the particle in terms of \(x\), and hence find the magnitude of the resisting force when \(x = 1\).
2. Find an expression for \(x\) in terms of \(t\).
3. Show that the particle is always less than 4 m from its initial position.

7 A particle \(P\) of mass 0.3 kg is projected vertically upwards from the ground with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at height \(x \mathrm {~m}\) above the ground, its upward speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that $$3 v - 90 \ln ( v + 30 ) + x = A ,$$ where \(A\) is a constant.

1. Differentiate this equation with respect to \(x\) and hence show that the acceleration of the particle is \(- \frac { 1 } { 3 } ( v + 30 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find, in terms of \(v\), the resisting force acting on the particle.
3. Find the time taken for \(P\) to reach its maximum height.

1. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force of magnitude \(F\) newtons. The force acts along the \(x\)-axis in the direction of \(x\) increasing. When \(P\) is \(x\) metres from the origin \(O\), it is moving away from \(O\) with speed \(\sqrt { \left( 8 x ^ { \frac { 3 } { 2 } } - 4 \right) } \mathrm { ms } ^ { - 1 }\).

Find \(F\) when \(P\) is 4 m from \(O\).

1. In this question you must show all stages in your working. Solutions relying entirely on calculator technology are not acceptable.

A particle \(P\) is moving along the \(x\)-axis.
At time \(t\) seconds, where \(0 \leqslant t \leqslant \frac { 2 } { 3 } , P\) is \(x\) metres from the origin 0 and is moving with velocity \(\mathrm { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) in the positive x direction where $$v = ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }$$ When \(\mathrm { t } = 0 , \mathrm { P }\) passes through 0 .

1. Find the value of x when the acceleration of P is \(243 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find v in terms of t .

1. A particle \(P\) of mass 0.5 kg moves on the \(x\)-axis under the action of a single force.

At time \(t\) seconds, \(t \geqslant 0\)

• \(O P = x\) metres, \(0 \leqslant x < \frac { \pi } { 2 }\)
• the force has magnitude \(\sin 2 x \mathrm {~N}\) and is directed towards the origin \(O\)
• \(P\) is moving in the positive \(x\) direction with speed \(v \mathrm {~ms} ^ { - 1 }\)

At time \(t = 0 , P\) passes through the origin with speed \(2 \mathrm {~ms} ^ { - 1 }\)

1. Show that \(v = 2 \cos x\)
2. Show that \(t = \frac { 1 } { 2 } \ln ( \sqrt { 2 } + 1 )\) when \(x = \frac { \pi } { 4 }\)

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} A particle \(P\) is moving along a straight line. At time \(t\) seconds, \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 50 } { 2 x + 3 } \mathrm {~ms} ^ { - 1 }\)

1. Find the deceleration of \(P\) when \(x = 12\) Given that \(x = 4\) when \(t = 1\)
2. find the value of \(t\) when \(x = 12\)

2. In this question solutions relying on calculator technology are not acceptable. A particle \(P\) of mass 2 kg is moving along the positive \(x\)-axis.
At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v \mathrm {~ms} ^ { - 1 }\) where \(v = \frac { 1 } { \sqrt { ( 2 x + 1 ) } }\)

1. Find the magnitude of the resultant force acting on \(P\) when its speed is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 1 }\) When \(t = 0 , P\) is at \(O\)
2. Find the value of \(t\) when \(P\) is 7.5 m from \(O\)

4. Whilst in free-fall a parachutist falls vertically such that his velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when he is \(x\) metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where \(k\) is a constant.
Given that he experiences an acceleration of \(\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }\),

1. show that \(f = g \mathrm { e } ^ { - \frac { 2 x } { k } }\). After falling a large distance, his velocity is constant at \(49 \mathrm {~ms} ^ { - 1 }\).
2. Find the value of \(k\).
3. Hence, express \(f\) in the form ( \(\lambda - \mu v ^ { 2 }\) ) where \(\lambda\) and \(\mu\) are constants which you should find.
   (4 marks)

7. A particle is travelling along the \(x\)-axis. At time \(t = 0\), the particle is at \(O\) and it travels such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a distance \(x\) metres from \(O\) is given by $$v = \frac { 2 } { x + 1 }$$ The acceleration of the particle is \(a \mathrm {~ms} ^ { - 2 }\).

1. Show that \(a = \frac { - 4 } { ( x + 1 ) ^ { 3 } }\).
   (4 marks) The points \(A\) and \(B\) lie on the \(x\)-axis. Given that the particle travels \(d\) metres from \(O\) to \(A\) in \(T\) seconds and 4 metres from \(A\) to \(B\) in 9 seconds,
2. show that \(d = 1.5\),
3. find \(T\).

4. A particle \(P\) of mass 0.5 kg moves in a horizontal straight line. At time \(t\) seconds \(( t \geqslant 0 )\), the displacement of \(P\) from a fixed point \(O\) of the line is \(x\) metres, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is moving in the direction of \(x\) increasing. A force of magnitude \(k x\) newtons acts on \(P\) in the direction \(P O\). The motion of \(P\) is also subject to a resistance of magnitude \(\lambda v\) newtons. Given that $$x = ( 1.5 + 10 t ) \mathrm { e } ^ { - 4 t }$$ find

1. the value of \(k\) and the value of \(\lambda\),
2. the distance from \(P\) to \(O\) when \(P\) is instantaneously at rest.

6. A particle of mass \(m\) is projected vertically upwards in a resisting medium. As the particle moves upwards, the speed \(v\) of the particle is given by $$v ^ { 2 } = k g \left( 5 \mathrm { e } ^ { - \frac { x } { 2 k } } - 4 \right)$$ where \(x\) is the distance of the particle above the point of projection and \(k\) is a positive constant.

1. Show that the magnitude of the resistance to the motion of the particle is \(\frac { m v ^ { 2 } } { 4 k }\).
   (4)
2. Find, in terms of \(k\), the greatest height reached by the particle above the point of projection.
3. Show that the time taken by the particle to reach its greatest height above the point of projection is \(\sqrt { \frac { 4 k } { g } } \arctan \left( \frac { 1 } { 2 } \right)\)

1. A particle is moving along the \(x\)-axis. At time \(t\) seconds the particle is \(x\) metres from the origin, \(O\), and its velocity \(v \mathrm {~ms} ^ { - 1 }\) is given by

$$v = \frac { 24 } { 4 x + 9 }$$

1. Find, in terms of \(x\), an expression for the acceleration of the particle at time \(t \mathrm {~s}\).
2. At \(t = T\) the acceleration of the particle is \(- \frac { 4 } { 3 } \mathrm {~ms} ^ { - 2 }\).
   1. Determine the value of \(x\) when \(t = T\).
   2. Given that \(x = - 2\) when \(t = 0\), find an expression for \(t\) in terms of \(x\) and hence find the value of \(T\).

A particle \(P\) of mass \(2\text{kg}\) moving on a horizontal straight line has displacement \(x\text{m}\) from a fixed point \(O\) on the line and velocity \(v\text{ms}^{-1}\) at time \(t\). The only horizontal force acting on \(P\) is a variable force \(F\text{N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass \(0.25\) kg is moving along the positive \(x\)-axis under the action of a single force. At time \(t\) seconds \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v\) m s\(^{-1}\) where \(\frac{\mathrm{d}v}{\mathrm{d}x} = 3\). It is given that \(x = 2\) and \(v = 3\) when \(t = 0\)

1. Find the magnitude of the force acting on \(P\) when \(x = 5\) [4]
2. Find the value of \(t\) when \(x = 5\) [4]

A particle \(P\) of mass \(m\) kg moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude \(m f(x)\) N, where \(f(x)\) is a function of the height \(x\) m of \(P\) above the ground. When \(P\) is at this height, its upward speed \(v\) ms\(^{-1}\) is given by $$v^2 = 2e^{-2gx} - 1.$$

1. Write down a differential equation for the motion of \(P\) and hence determine \(f(x)\) in terms of \(g\) and \(x\). [5 marks]
2. Show that the greatest height reached by \(P\) above the ground is \(\frac{1}{2g} \ln 2\) m. [2 marks]

Given that the work, in J, done by \(P\) against the resisting force as it moves from ground level to a point \(H\) m above the ground is equal to \(\int_0^H m f(x) dx\),

1. show that the total work done by \(P\) against the resistance during its upward motion is \(\frac{1}{2}m(1 - \ln 2)\) J. [3 marks]

A particle P of mass 4 kilograms moves in such a way that its position vector at time \(t\) seconds is \(\mathbf{r}\) metres, where $$\mathbf{r} = 3t\mathbf{i} + 2e^{-3t}\mathbf{j}.$$

1. Find the initial kinetic energy of P. [4]
2. Find the time when the acceleration of P is 2 metres per second squared. [3]