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

2. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the positive \(x\)-direction. The only force on \(P\) is a force of magnitude \(\left( 2 t + \frac { 1 } { 2 } \right) \mathrm { N }\) acting in the direction of \(x\) increasing, 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 \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance \(O A\).

2. A particle \(P\) of mass \(m\) is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When \(P\) is at a distance \(x\) from the centre of the Earth, the gravitational force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant. 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 \(k = m g R ^ { 2 }\). When \(P\) is at a height \(\frac { R } { 4 }\) above the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 2 } }\) Given that air resistance can be ignored,
2. find, in terms of \(R\), the greatest distance from the centre of the Earth reached by \(P\).

1. A particle \(P\) moves on the positive \(x\)-axis. When the displacement of \(P\) from \(O\) is \(x\) metres, its acceleration is \(( 6 - 4 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the direction of \(x\) increasing. Initially \(P\) is at \(O\) and the velocity of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O x\).

Find the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest.

1 A uniform disc with centre \(O\) has mass \(m\) and radius \(a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(O\). One end of a light inextensible string is attached to a point on the circumference and is wrapped several times round the circumference. A particle \(P\), of mass \(2 m\), is attached to the free end of the string and the disc is held at rest with \(P\) hanging freely. The system is released from rest. Assuming that resistances may be neglected, find the acceleration of \(P\).

4 A particle \(P\) of mass \(m\) moves along part of a horizontal straight line \(A B\). The mid-point of \(A B\) is \(O\), and \(A B = 4 a\). At time \(t , A P = 2 a + x\). The particle \(P\) is acted on by two horizontal forces. One force has magnitude \(m g \left( \frac { 2 a + x } { 2 a } \right) ^ { - \frac { 1 } { 2 } }\) and acts towards \(A\); the other force has magnitude \(m g \left( \frac { 2 a - x } { 2 a } \right)\) and acts towards \(B\). Show that, provided \(\frac { x } { a }\) remains small, \(P\) moves in approximate simple harmonic motion with centre \(O\), and state the period of this motion. At time \(t = 0 , P\) is released from rest at the point where \(x = \frac { 1 } { 20 } a\). Show that the speed of \(P\) when \(x = \frac { 1 } { 40 } a\) is \(\frac { 1 } { 80 } \sqrt { } ( 3 a g )\), and find the value of \(t\) when \(P\) reaches this point for the first time.

5 A particle of mass \(m\) moves in a straight line \(A B\) of length \(2 a\). When the particle is at a general point \(P\) there are two forces acting, one in the direction \(\overrightarrow { P A }\) with magnitude \(m g \left( \frac { P A } { a } \right) ^ { - \frac { 1 } { 4 } }\) and the other in the direction \(\overrightarrow { P B }\) with magnitude \(m g \left( \frac { P B } { a } \right) ^ { \frac { 1 } { 2 } }\). At time \(t = 0\) the particle is released from rest at the point \(C\), where \(A C = 1.04 a\). At time \(t\) the distance \(A P\) is \(a + x\). Show that the particle moves in approximate simple harmonic motion. Using the approximate simple harmonic motion, find the speed of \(P\) when it first reaches the mid-point of \(A B\) and the time taken for \(P\) to first reach half of this speed.

5 A car of mass 1500 kg travels up a line of greatest slope of a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The power of the car's engine is constant and equal to 25 kW and the resistance to the motion of the car is constant and equal to 750 N . The car passes through point \(A\) with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the car at \(A\). The car later passes through a point \(B\) with speed \(20 \mathrm {~ms} ^ { - 1 }\). The car takes 28s to travel from \(A\) to \(B\).
2. Find the distance \(A B\).

5 The engine of a car of mass 1200 kg produces a maximum power of 40 kW .
In an initial model of the motion of the car the total resistance to motion is assumed to be constant.

1. Given that the greatest steady speed of the car on a straight horizontal road is \(42 \mathrm {~ms} ^ { - 1 }\), find the magnitude of the resistance force. The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
2. Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. (a) Find the power of the engine of the car at this instant.
   (b) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
4. Without carrying out any further calculations,
   (a) explain whether the time taken to attain a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
   (b) explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.

3 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~ms} ^ { - 1 }\) along a straight horizontal road using maximum power, its acceleration is \(3.3 \mathrm {~ms} ^ { - 2 }\). In an initial model of the motion of the car it is assumed that the resistance to motion is constant.

1. Using this initial model, find the greatest possible steady speed of the car along the road. In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
2. Using this refined model, find the greatest possible steady speed of the car along the road. The greatest possible steady speed of the car on the road is measured and found to be \(21.6 \mathrm {~ms} ^ { - 1 }\).
3. Explain what this value means about the models used in parts (a) and (b).

2 A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW . In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude \(R N\). The car is initially driven on a straight horizontal road. At the instant that its speed is \(20 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 430\).
2. Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be 60 v where \(v\) is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle \(\alpha\) above the horizontal where \(\sin \alpha = 0.2\).
3. Determine the speed of the car at the instant that its acceleration is \(0.15 \mathrm {~ms} ^ { - 2 }\) up the slope, according to the revised model.

7 An engineer is modelling the motion of a particle \(P\) of mass 0.5 kg in a wind tunnel. \(P\) is modelled as travelling in a straight line. The point \(O\) is a fixed point within the wind tunnel. The displacement of \(P\) from \(O\) at time \(t\) seconds is \(x\) metres, for \(t \geqslant 0\). You are given that \(x \geqslant 0\) for all \(t \geqslant 0\) and that \(P\) does not reach the end of the wind tunnel.
If \(t \geqslant 0\), then \(P\) is subject to three forces which are modelled in the following way.

• The first force has a magnitude of \(5 ( t + 1 ) \cosh t \mathrm {~N}\) and acts in the positive \(x\)-direction.
• The second force has a magnitude of \(0.5 x \mathrm {~N}\) and acts towards \(O\).
• The third force has a magnitude of \(\left| \frac { d x } { d t } \right| \mathrm { N }\) and acts in the direction of motion of the particle.
  1. 1. Show that the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t\) is \(6 - 5 t\).
     2. Use your answer to part (a)(ii) to show that the term in \(t ^ { 2 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(- 3 t ^ { 2 }\).
     3. By differentiating the differential equation in part (a) with respect to \(t\), show that the term in \(t ^ { 3 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(0.5 t ^ { 3 }\). You are given that the complete Maclaurin series for the function f is valid for all values of \(t \geqslant 0\).
        After 0.25 seconds \(P\) has travelled 1.43 m towards the origin.
  2. 1. By using the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\), evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 0.25\).
     2. Explain why it might not be sensible to use the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\) to evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 10\).

5 A particle \(P\) of mass 4.5 kg is free to move along the \(x\)-axis. In a model of the motion it is assumed that \(P\) is acted on by two forces:

• a constant force of magnitude \(f \mathrm {~N}\) in the positive \(x\) direction;
• a resistance to motion, \(R \mathrm {~N}\), whose magnitude is proportional to the speed of \(P\).

At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at the origin \(O\) and is moving in the positive direction with speed \(u \mathrm {~ms} ^ { - 1 }\), and when \(v = 5 , R = 2\). \begin{enumerate}[label=(\alph*)] \item Show that, according to the model, \(\frac { d v } { d t } = \frac { 10 f - 4 v } { 45 }\). \item

1. By solving the differential equation in part (a), show that \(\mathrm { v } = \frac { 1 } { 2 } \left( 5 \mathrm { f } - ( 5 \mathrm { f } - 2 \mathrm { u } ) \mathrm { e } ^ { - \frac { 4 } { 45 } \mathrm { t } } \right)\).
2. Describe briefly how, according to the model, the speed of \(P\) varies over time in each of the following cases.

6 A particle \(P\) of mass 2.5 kg is free to move along the \(x\)-axis. When its displacement from the origin is \(x \mathrm {~m}\) its velocity is \(v \mathrm {~ms} ^ { - 1 }\). At time \(t = 0\) seconds, \(P\) is at the point where \(x = 1\) and is travelling in the negative \(x\)-direction with speed \(5 \mathrm {~ms} ^ { - 1 }\). At this time an impulse of \(I\) Ns is applied to \(P\) in the positive \(x\)-direction so that \(P\) moves in the positive \(x\)-direction with speed \(18 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(I\). Subsequently, whenever \(P\) is in motion, two forces act on it. The first force acts in the positive \(x\)-direction and has magnitude \(\frac { 5 v ^ { 2 } } { x } N\). The second force acts in the negative \(x\)-direction and has magnitude 60 vN .
2. Show that the motion of \(P\) can be modelled by the differential equation \(\frac { \mathrm { dV } } { \mathrm { dx } } = \frac { \mathrm { aV } } { \mathrm { x } } + \mathrm { b }\) where \(a\) and \(b\) are constants whose values should be determined.
3. By solving the differential equation derived in part (b) find an expression for \(v\) in terms of \(x\). You are given that \(\mathrm { x } = \frac { 4 } { 3 \mathrm { e } ^ { - 24 \mathrm { t } } + 1 }\) when \(t \geqslant 0\).
4. Describe in detail the motion of \(P\) when \(t \geqslant 0\).

3 A particle \(P\) of mass \(m\) moves on the \(x\)-axis under the action of a force \(F\) directed along the axis. When the displacement of \(P\) from the origin is \(x\) its velocity is \(v\).

1. By using the fact that the dimensions of the derivative \(\frac { d v } { d x }\) are the same as those of \(\frac { v } { x }\), verify that the equation \(\mathrm { F } = \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dx } }\) is dimensionally consistent. It is given that \(\mathrm { v } = \mathrm { km } ^ { - \frac { 1 } { 2 } } \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } }\) where \(a\) and \(k\) are constants.
2. Explain why \([ a ]\) must be the same as \([ x ]\).
3. Deduce the dimensions of \(k\).
4. Find an expression for \(F\) in terms of \(x\) and \(k\).

5 A particle \(P\) of mass 3 kg moves on the \(x\)-axis under the action of a single force acting in the positive \(x\)-direction. At time \(t \mathrm {~s}\), where \(t \geqslant 0\), the displacement of \(P\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the force acting is inversely proportional to \(( t + 1 ) ^ { 2 }\). Initially \(P\) is at rest at the point where \(x = 1\). When \(t = 1 , v = 2\).

1. Show that \(\frac { \mathrm { dv } } { \mathrm { dt } } = \frac { \mathrm { k } } { 3 ( \mathrm { t } + 1 ) ^ { 2 } }\) where \(k\) is a constant.
2. Find an expression for \(v\) in terms of \(t\).
3. Find an expression for \(x\) in terms of \(t\). As \(t\) increases, \(v\) approaches a limiting value, \(\mathrm { V } _ { \mathrm { T } }\).
4. Determine how far \(P\) is from its initial position at the instant when \(v\) is \(95 \%\) of \(\mathrm { V } _ { \mathrm { T } }\).

5 A particle \(P\) of mass \(m \mathrm {~kg}\) is projected vertically upwards through a liquid. Student \(A\) measures \(P\) 's initial speed as \(( 8.5 \pm 0.25 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and they also record the time for \(P\) to attain its greatest height above the initial point of projection as over 3 seconds.
In an attempt to model the motion of \(P\) student \(B\) determines that \(t\) seconds after projection the only forces acting on \(P\) are its weight and the resistance from the liquid. Student \(B\) models the resistance from the liquid to be of magnitude \(m v ^ { 2 } - 6 m v\), where \(v\) is the speed of the particle.

1. (a) Show that \(\frac { \mathrm { d } t } { \mathrm {~d} v } = - \frac { 1 } { ( v - 3 ) ^ { 2 } + 0.8 }\).
   (b) Determine whether student \(B\) 's model is consistent with the time recorded by \(A\) for \(P\) to attain its greatest height. After attaining its greatest height \(P\) now falls through the liquid. Student \(C\) claims that the time taken for \(P\) to achieve a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when falling through the liquid is given by $$- \int _ { 0 } ^ { 1 } \frac { 1 } { ( v - 3 ) ^ { 2 } + 0.8 } \mathrm {~d} v$$
2. Explain why student \(C\) 's claim is incorrect and write down the integral which would give the correct time for \(P\) to achieve a speed of \(1 \mathrm {~ms} ^ { - 1 }\) when falling through the liquid.

5 A sphere of mass 200 grams is released from rest and allowed to fall vertically.

1. A student states that the acceleration of the sphere is \(9.8 \mathrm {~ms} ^ { - 2 }\) while it is falling. What modelling assumption is this student making?
2. The student conducts an experiment and finds that the acceleration of the ball is in fact \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). He formulates a model for the motion that assumes a constant resistance force acts on the ball as it is falling.
   1. Calculate the magnitude of this resistance force based on this assumption.
   2. Describe how the resistance force would vary in reality.
3. In a revised model the resistance force is assumed to be proportional to the speed of the sphere.
   1. State the initial acceleration of the sphere.
   2. State what would happen to the acceleration of the sphere if it were able to fall for a long period of time.

6 A student is modelling the motion of a small boat as it moves on a lake. When the speed of the boat is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. At time \(t\) seconds later, it has a velocity of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and experiences a resistance force of magnitude \(20 v\) newtons. The mass of the boat is 80 kg . To set up a simple model for the motion of the boat, the student assumes that the water in the lake is still and that the boat travels in a straight line.

1. Explain how these two assumptions allow the student to create a simple model.
2. State one other assumption that the student should make.
3. 1. Express \(\frac { \mathrm { d } v } { \mathrm {~d} t }\) in terms of \(v\).
   2. Find an expression for \(v\) in terms of \(t\).

8 Vicky has mass 65 kg and is skydiving. She steps out of a helicopter and falls vertically. She then waits a short period of time before opening her parachute. The parachute opens at time \(t = 0\) when her speed is \(19.6 \mathrm {~ms} ^ { - 1 }\), and she then experiences an air resistance force of magnitude \(260 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is her speed at time \(t\) seconds.

1. When \(t > 0\) :
   1. show that the resultant downward force acting on Vicky is 65(9.8-4v) newtons
   2. show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 4 ( v - 2.45 )\).
2. By showing that \(\int \frac { 1 } { v - 2.45 } \mathrm {~d} v = - \int 4 \mathrm {~d} t\), find \(v\) in terms of \(t\).
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6 Alice places a toy, of mass 0.4 kg , on a slope. The toy is set in motion with an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) down the slope. The resultant force acting on the toy is \(( 2 - 4 v )\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the toy's velocity at time \(t\) seconds after it is set in motion.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 10 ( v - 0.5 )\).
2. By using \(\int \frac { 1 } { v - 0.5 } \mathrm {~d} v = - \int 10 \mathrm {~d} t\), find \(v\) in terms of \(t\).
3. Find the time taken for the toy's velocity to reduce to \(0.55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(7 \quad\) A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). With the string taut, the bead is at the point \(B\), vertically below \(O\), when it is set into vertical circular motion with an initial horizontal velocity \(u\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-5_616_613_520_733} The string does not become slack in the subsequent motion. The velocity of the bead at the point \(A\), where \(A\) is vertically above \(O\), is \(v\).

5 A particle, of mass 12 kg , is moving along a straight horizontal line. At time \(t\) seconds, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the particle moves, it experiences a resistance force of magnitude \(4 v ^ { \frac { 1 } { 3 } }\). No other horizontal force acts on the particle. The initial speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$v = \left( 4 - \frac { 2 } { 9 } t \right) ^ { \frac { 3 } { 2 } }$$
2. Find the value of \(t\) when the particle comes to rest.

7 A particle of mass 20 kg moves along a straight horizontal line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance force of magnitude \(10 \sqrt { v }\) newtons acts on the particle while it is moving. At time \(t = 0\) the velocity of the particle is \(25 \mathrm {~ms} ^ { - 1 }\).

1. Show that, at time \(t\) $$v = \left( \frac { 20 - t } { 4 } \right) ^ { 2 }$$
2. State the value of \(t\) when the particle comes to rest.