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

3 A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement is \(x \mathrm {~m}\) from \(O\). The force acting on \(B\) has magnitude \(\left( 2 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and is directed horizontally away from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 8\).
2. Find the velocity of \(B\) when \(x = 1.5\). An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 6 - 3 x$$
3. Find the magnitude of this extra force and state the direction in which it acts.

5 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude \(3 \mathrm { e } ^ { - t } \mathrm {~N}\) directed up a line of greatest slope acts on \(P\), where \(t \mathrm {~s}\) is the time after release.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 7.5 \mathrm { e } ^ { - t } - 5\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) up the plane at time \(t \mathrm {~s}\).
2. Express \(v\) in terms of \(t\).
3. Find the distance of \(P\) from \(O\) when \(v\) has its maximum value.

7 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, and travels down a line of greatest slope. The coefficient of friction between \(P\) and the plane is 0.3 . A force of magnitude \(0.6 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \sqrt { } 3 - 1.5 - 3 x\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at a displacement \(x \mathrm {~m}\) from \(O\).
2. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
3. Calculate the magnitude of the acceleration of \(P\) immediately after it has first come to instantaneous rest.

1 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(t \mathrm { e } ^ { - v } \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after release. Find the velocity of \(P\) when \(t = 2\).

3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves down the line of greatest slope through \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement from \(O\) is \(x \mathrm {~m}\). A retarding force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 0.5 v ^ { 2 }\).
2. Express \(v\) in terms of \(x\).

3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). After projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(8 x \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude ( \(2 \mathrm { e } ^ { - x } + 4\) ) N opposes the motion of \(P\). One end of a light elastic string of natural length 0.5 m is attached to \(O\) and the other end of the string is attached to \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 20 x - 10 - 5 \mathrm { e } ^ { - x }\) before the elastic string becomes taut.
2. Given that the initial velocity of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(v\) when the string first becomes taut.
   When the string is taut, the acceleration of \(P\) is proportional to \(\mathrm { e } ^ { - x }\).
3. Find the modulus of elasticity of the string.

3 A smooth horizontal surface has two fixed points \(O\) and \(A\) which are 0.8 m apart. A particle \(P\) of mass 0.25 kg is projected with velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally from \(A\) in the direction away from \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(k v ^ { 2 } x ^ { - 2 } \mathrm {~N}\) opposes the motion of \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 k v ^ { 2 } x ^ { - 2 }\).
2. Express \(v\) in terms of \(k\) and \(x\).

6 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\) on a smooth horizontal surface. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A horizontal force of variable magnitude \(0.09 \sqrt { } x \mathrm {~N}\) directed away from \(O\) acts on \(P\). An additional force of constant magnitude 0.3 N directed towards \(O\) acts on \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 0.45 \sqrt { } x - 1.5\).
2. Find the value of \(x\) for which the acceleration of \(P\) is zero.
3. Given that the minimum value of \(v\) is positive, find the set of possible values for the speed of projection.

5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
2. Find the greatest height of \(P\) above the surface.
3. Find the speed of \(P\) immediately before it strikes the surface after descending.

4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude \(2 v ^ { 2 } \mathrm {~N}\). The initial velocity of the particle is \(3 \mathrm {~ms} ^ { - 1 }\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce the limiting value of \(v\).

2 A ball of mass 2 kg is projected vertically downwards with speed \(5 \mathrm {~ms} ^ { - 1 }\) through a liquid. At time \(t \mathrm {~s}\) after projection, the velocity of the ball is \(v \mathrm {~ms} ^ { - 1 }\) and its displacement from its starting point is \(x \mathrm {~m}\). The forces acting on the ball are its weight and a resistive force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce what happens to \(v\) for large values of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side \(2 a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(A B\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(E P\) is perpendicular to the side \(A B\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac { 1 } { 2 }\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan \theta\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length \(8 a\) and modulus of elasticity \(5 m g\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12 a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(A P = B P = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(A B\) it has speed \(\sqrt { 80 a g }\).
   1. Find \(L\) in terms of \(a\).
   2. Find the initial acceleration of \(P\) in terms of \(g\).

4. The resistance to the motion of a cyclist is modelled as \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\), at a constant speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 4\). The cyclist ascends a slope inclined at an angle \(\beta\) to the horizontal, where \(\sin \beta = \frac { 1 } { 40 }\), at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate at which the cyclist is working.
   (6 marks)

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. A particle \(P\) of mass 3 kg is moving along the horizontal \(x\)-axis. At time \(t = 0 , P\) passes through the origin \(O\) moving in the positive \(x\) direction. At time \(t\) seconds, \(O P = x\) metres and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds, the resultant force acting on \(P\) is \(\frac { 9 } { 2 } ( 26 - x ) \mathrm { N }\), measured in the positive \(x\) direction. For \(t > 0\) the maximum speed of \(P\) is \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find \(v ^ { 2 }\) in terms of \(x\).

2. A particle \(P\) of mass 0.6 kg is moving along the positive \(x\)-axis in the positive direction. The only force acting on \(P\) acts in the direction of \(x\) increasing and has magnitude \(\left( 3 t + \frac { 1 } { 2 } \right) \mathrm { N }\), where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0 , P\) is at rest at \(O\).

1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(\frac { 10 } { 3 } \mathrm {~ms} ^ { - 1 }\).
2. Find the distance \(O A\).

2. A particle \(P\) of mass \(m\) is at a distance \(x\) above the surface of the Earth. The Earth exerts a gravitational force on \(P\). This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of \(P\) from the centre of the Earth. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }\) A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance \(R\) above the surface of the Earth, the particle has speed \(U\). Air resistance is modelled as being negligible.
2. Find, in terms of \(U , g\) and \(R\), the speed of the particle when it strikes the surface of the Earth.
   

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1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\)

The acceleration of \(P\) has magnitude \(\frac { 2 } { ( 2 x + 1 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\) When \(t = 0 , P\) passes through \(O\) in the positive \(x\) direction with speed \(1 \mathrm {~ms} ^ { - 1 }\)

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

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(O\) on a rough plane which is inclined to the horizontal at an angle \(\alpha\) The string lies along a line of greatest slope of the plane.
The particle \(P\) is held at rest on the plane at the point \(A\), where \(O A = a\), as shown in Figure 5. The particle \(P\) is released from \(A\) and slides down the plane, coming to rest at the point \(B\). The coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \tan \alpha\) Air resistance is modelled as being negligible.

1. Show that \(A B = a ( \sin \alpha - \mu \cos \alpha )\). Given that \(\tan \alpha = \frac { 3 } { 4 }\) and \(\mu = \frac { 1 } { 2 }\)
2. find, in terms of \(a\) and \(g\), the maximum speed of \(P\) as it moves from \(A\) to \(B\)
3. Describe the motion of \(P\) after it reaches \(B\), justifying your answer.

1. A particle \(P\) of mass \(m \mathrm {~kg}\) is initially held at rest at the point \(O\) on a smooth inclined plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\)

The particle is released from rest and slides down the plane against a force which acts towards \(O\). The force has magnitude \(\frac { 1 } { 3 } m x ^ { 2 } \mathrm {~N}\), where \(x\) metres is the distance of \(P\) from \(O\).

1. Find the speed of \(P\) when \(x = 2\) The particle first comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

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 spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre, \(O\), of a planet.

The planet is modelled as a fixed sphere of radius \(R\).
The spacecraft \(S\) is modelled as a particle.
The gravitational force of the planet is the only force acting on \(S\).
When \(S\) is a distance \(x ( x \geqslant R )\) from \(O\)

• the gravitational force is directed towards \(O\) and has magnitude \(\frac { m g R ^ { 2 } } { 2 x ^ { 2 } }\)
• the speed of \(S\) is \(v\)
  1. Show that

$$v ^ { 2 } = \frac { g R ^ { 2 } } { x } + C$$ where \(C\) is a constant. When \(x = 3 R , v = \sqrt { 3 g R }\)

Find, in terms of \(g\) and \(R\), the speed of \(S\) as it hits the surface of the planet.

2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.

1. Show that \(K = m g R ^ { 2 }\) (2) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.