6.06a Variable force: dv/dt or v*dv/dx methods

5. A particle \(P\) of mass 0.4 kg moves on the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { x ^ { 2 } }\) newtons, where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 2\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(x = 5\) the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(k\). The particle first comes to instantaneous rest at the point \(A\).
2. Find the value of \(x\) at \(A\).

1. A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis in the positive \(x\) direction under the action of a resultant force. This force acts in the direction of \(x\) increasing. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O , P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the force has magnitude \(\frac { 4 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\).

When \(t = 0 , P\) is at rest at \(O\).

1. Show that \(v ^ { 2 } = 5 \left( \frac { ( x + 1 ) ^ { 2 } - 1 } { ( x + 1 ) ^ { 2 } } \right)\) When \(t = 2 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
2. Using algebraic integration, find the distance \(A B\).

5. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 1 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 2 } { x ^ { 3 } } \mathrm {~N}\) and is directed towards \(O\). When \(t = 1 , x = 1\) and \(v = 3\) Show that

1. \(v ^ { 2 } = \frac { 4 } { x ^ { 2 } } + 5\)
2. \(t = \frac { a + \sqrt { b x ^ { 2 } + c } } { d }\), where \(a , b , c\) and \(d\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-13_2255_50_314_34}
   

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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 \(x = 12\)

1. The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
   At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.

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

• \(\quad P\) is a distance \(x\) from \(O\)
• \(P\) is moving with speed \(v\)
• \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)

Air resistance is modelled as being negligible.

1. Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\) Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
2. Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
3. Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.

1. A particle \(P\) is moving along the \(x\)-axis.

At time \(t\) seconds, where \(t \geqslant 0\), the displacement of \(P\) from the origin \(O\) is \(x\) metres and \(P\) is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) is \(\frac { 3 \sqrt { x + 1 } } { 4 } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.
When \(t = 0 , x = 15\) and \(v = 8\)

1. Show that \(v = ( x + 1 ) ^ { \frac { 3 } { 4 } }\)
2. Find \(t\) in terms of \(v\).

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 \(O\)

3. At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) has magnitude \(2 ( t + 4 ) ^ { - \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\).

1. Show that, at time \(t\) seconds, the velocity of \(P\) is \(16 - 4 ( t + 4 ) ^ { \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 1 }\)
2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.

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5. A car of mass 800 kg moves along a horizontal straight road. At time \(t\) seconds, the resultant force acting on the car has magnitude \(\frac { 48000 } { ( t + 2 ) ^ { 2 } }\) newtons, acting in the direction of the motion of the car. When \(t = 0\), the car is at rest.

1. Show that the speed of the car approaches a limiting value as \(t\) increases and find this value.
2. Find the distance moved by the car in the first 6 seconds of its motion.

2. A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal table. When \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line, it experiences a force of magnitude \(\frac { 16 } { 5 x ^ { 2 } } \mathrm {~N}\) away from \(O\) in the direction \(O P\). Initially \(P\) is at a point 2 m from \(O\) and is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when \(P\) first comes to rest.

1. A particle \(P\) of mass 3 kg is moving in a straight line. At time \(t\) seconds, \(0 \leqslant t \leqslant 4\), the only force acting on \(P\) is a resistance to motion of magnitude \(\left( 9 + \frac { 15 } { ( t + 1 ) ^ { 2 } } \right) \mathrm { N }\). At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 4 , v = 0\).

Find the value of \(v\) when \(t = 0\).

1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis. At time \(t\) seconds, \(P\) is moving under the action of a single force of magnitude \([ 4 + \cos ( \pi t ) ] \mathrm { N }\), directed away from the origin. When \(t = 1\), the particle \(P\) is moving away from the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find the speed of \(P\) when \(t = 1.5\), giving your answer to 3 significant figures.

3. A particle \(P\) is moving in a straight line. At time \(t\) seconds, \(P\) is at a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 10 } { x + 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) when \(x = 14\) Given that \(x = 2\) when \(t = 1\),
2. find the value of \(t\) when \(x = 14\)

1. A particle \(P\) is moving along the positive \(x\)-axis. When the displacement of \(P\) from the origin is \(x\) metres, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of \(P\) is \(9 x \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

When \(x = 2 , v = 6\) Show that \(v ^ { 2 } = 9 x ^ { 2 }\).
(4)

3. A particle \(P\) of mass \(m \mathrm {~kg}\) slides from rest down a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When \(P\) has moved a distance \(x\) metres down the plane, the resistance to the motion of \(P\) from non-gravitational forces has magnitude \(m x ^ { 2 }\) newtons. Find

1. the speed of \(P\) when \(x = 2\),
2. the distance \(P\) has moved when it comes to rest for the first time.

7. A particle \(P\) of mass \(\frac { 1 } { 3 } \mathrm {~kg}\) moves along the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { ( x + 1 ) ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 1\) the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(x = 8\) the speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
   (Total 14 marks)

1. A particle \(P\) of mass 0.5 kg moves on the positive \(x\)-axis under the action of a single force directed towards the origin \(O\). At time \(t\) seconds the distance of \(P\) from \(O\) is \(x\) metres, the magnitude of the force is \(0.375 x ^ { 2 } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

When \(t = 0 , O P = 8 \mathrm {~m}\) and \(P\) is moving towards \(O\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 260 - \frac { 1 } { 2 } \chi ^ { 3 }\).
2. Find the distance of \(P\) from \(O\) at the instant when \(v = 5\).

2. A particle of mass 4 kg is moving along the horizontal \(x\)-axis under the action of a single force which acts in the positive \(x\)-direction. At time \(t\) seconds the force has magnitude \(\left( 1 + 3 t ^ { \frac { 1 } { 2 } } \right) \mathrm { N }\).
When \(t = 0\) the particle has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the work done by the force in the interval \(0 \leqslant t \leqslant 4\)