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

2 A particle of mass \(m \mathrm {~kg}\) moves horizontally in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The total resistance force on the particle is of magnitude \(m k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\) where \(k\) is a positive constant. There are no other horizontal forces present. Initially \(v = 25\) and the particle is at a point O .

1. Show that \(v = 4 \left( k t + \frac { 2 } { 5 } \right) ^ { - 2 }\).
2. Find the displacement from O of the particle at time \(t\).
3. Describe the motion of the particle as \(t\) increases. Section B (48 marks)

1 A raindrop of mass \(m\) falls vertically from rest under gravity. Initially the mass of the raindrop is \(m _ { 0 }\). As it falls it loses mass by evaporation at a rate \(\lambda m\), where \(\lambda\) is a constant. Its motion is modelled by assuming that the evaporation produces no resultant force on the raindrop. The velocity of the raindrop is \(v\) at time \(t\). The forces on the raindrop are its weight and a resistance force of magnitude \(k m v\), where \(k\) is a constant.

1. Find \(m\) in terms of \(m _ { 0 } , \lambda\) and \(t\).
2. Write down the equation of motion of the raindrop. Solve this differential equation and hence show that \(v = \frac { g } { \lambda - k } \left( \mathrm { e } ^ { ( \lambda - k ) t } - 1 \right)\).
3. Find the velocity of the raindrop when it has lost half of its initial mass.

3 A car of mass 800 kg moves horizontally in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds. While \(v \leqslant 20\), the power developed by the engine is \(8 v ^ { 4 } \mathrm {~W}\). The total resistance force on the car is of magnitude \(8 v ^ { 2 } \mathrm {~N}\). Initially \(v = 2\) and the car is at a point O . At time \(t\) s the displacement from O is \(x \mathrm {~m}\).

1. Find \(v\) in terms of \(x\) and show that when \(v = 20 , x = 100 \ln 1.9\).
2. Find the relationship between \(t\) and \(x\), and show that when \(v = 20 , t \approx 19.2\). The driving force is removed at the instant when \(v\) reaches 20 .
3. For the subsequent motion, find \(v\) in terms of \(t\). Calculate \(t\) when \(v = 2\).

1 A rocket in deep space has initial mass \(m _ { 0 }\) and is moving in a straight line at speed \(v _ { 0 }\). It fires its engine in the direction opposite to the motion in order to increase its speed. The propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) after the engines are fired, the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
2. Hence find an expression for \(v\) at time \(t\).

2 A light elastic string AB has stiffness \(k\). The end A is attached to a fixed point and a particle of mass \(m\) is attached at the end B . With the string vertical, the particle is released from rest from a point at a distance \(a\) below its equilibrium position. At time \(t\), the displacement of the particle below the equilibrium position is \(x\) and the velocity of the particle is \(v\).

1. Show that $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - k x$$
2. Show that, while the particle is moving upwards and the string is taut, $$v = - \sqrt { \frac { k } { m } \left( a ^ { 2 } - x ^ { 2 } \right) }$$
3. Hence use integration to find an expression for \(x\) at time \(t\) while the particle is moving upwards and the string is taut.

3 A model car of mass 2 kg moves from rest along a horizontal straight path. After time \(t \mathrm {~s}\), the velocity of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power, \(P \mathrm {~W}\), developed by the engine is initially modelled by \(P = 2 v ^ { 3 } + 4 v\). The car is subject to a resistance force of magnitude \(6 v \mathrm {~N}\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = ( 1 - v ) ( 2 - v )\) and hence show that \(t = \ln \frac { 2 - v } { 2 ( 1 - v ) }\).
2. Hence express \(v\) in terms of \(t\). Once the power reaches 4.224 W it remains at this constant value with the resistance force still acting.
3. Verify that the power of 4.224 W is reached when \(v = 0.8\) and calculate the value of \(t\) at this instant.
4. Find \(v\) in terms of \(t\) for the motion at constant power. Deduce the limiting value of \(v\) as \(t \rightarrow \infty\).

1 A sports car of mass 1.2 tonnes is being tested on a horizontal, straight section of road. After \(t \mathrm {~s}\), the car has travelled \(x \mathrm {~m}\) from the starting line and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine produces a driving force of 4000 N and the total resistance to the motion of the car is given by \(\frac { 40 } { 49 } v ^ { 2 } \mathrm {~N}\). The car crosses the starting line with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down an equation of motion for the car and solve it to show that \(v ^ { 2 } = 4900 - 4800 \mathrm { e } ^ { - \frac { 1 } { 735 } x }\).
2. Hence find the work done against the resistance to motion over the first 100 m beyond the starting line.

1 A car of mass \(m\) moves horizontally in a straight line. At time \(t\) the car is a distance \(x\) from a point O and is moving away from O with speed \(v\). There is a force of magnitude \(k v ^ { 2 }\), where \(k\) is a constant, resisting the motion of the car. The car's engine has a constant power \(P\). The terminal speed of the car is \(U\).

1. Show that $$m v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = P \left( 1 - \frac { v ^ { 3 } } { U ^ { 3 } } \right)$$
2. Show that the distance moved while the car accelerates from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\) is $$\frac { m U ^ { 3 } } { 3 P } \ln A$$ stating the exact value of the constant \(A\). Once the car attains a speed of \(\frac { 1 } { 2 } U\), no further power is supplied by the car's engine.
3. Find, in terms of \(m , P\) and \(U\), the time taken for the speed of the car to reduce from \(\frac { 1 } { 2 } U\) to \(\frac { 1 } { 4 } U\).

4 A raindrop falls from rest through a stationary cloud. The raindrop has mass \(m\) and speed \(v\) when it has fallen a distance \(x\). You may assume that resistances to motion are negligible.

1. Derive the equation of motion $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } + v ^ { 2 } \frac { \mathrm {~d} m } { \mathrm {~d} x } = m g .$$ Initially the mass of the raindrop is \(m _ { 0 }\). Two different models for the mass of the raindrop are suggested.
   In the first model \(m = m _ { 0 } \mathrm { e } ^ { k _ { 1 } x }\), where \(k _ { 1 }\) is a positive constant.
2. Show that $$v ^ { 2 } = \frac { g } { k _ { 1 } } \left( 1 - \mathrm { e } ^ { - 2 k _ { 1 } x } \right) ,$$ and hence state, in terms of \(g\) and \(k _ { 1 }\), the terminal velocity of the raindrop according to this first model. In the second model \(m = m _ { 0 } \left( 1 + k _ { 2 } x \right)\), where \(k _ { 2 }\) is a positive constant.
3. By considering the expression obtained from differentiating \(v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 }\) with respect to \(x\), show that, for the second model, the equation of motion in part (i) may be written as $$\frac { d } { d x } \left[ v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 } \right] = 2 g \left( 1 + k _ { 2 } x \right) ^ { 2 } .$$ Hence find an expression for \(v ^ { 2 }\) in terms of \(g , k _ { 2 }\) and \(x\).
4. Suppose that the models give the same value for the speed of the raindrop at the instant when it has doubled its initial mass. Find the exact value of \(\frac { k _ { 1 } } { k _ { 2 } }\), giving your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. are integers. \section*{END OF QUESTION PAPER}

6. The vertices of a tetrahedron \(P Q R S\) have position vectors \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and \(\mathbf { s }\) respectively, where $$\mathbf { p } = - 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad \mathbf { q } = 4 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \mathbf { s } = 4 \mathbf { i } + \mathbf { k }$$ Forces of magnitude 20 N and \(2 \sqrt { } 13 \mathrm {~N}\) act along \(S Q\) and \(S R\) respectively. A third force acts at \(P\).
Given that the system of three forces reduces to a couple \(\mathbf { G }\), find

1. the third force,
2. the magnitude of \(\mathbf { G }\).
   (6)
   (Total 12 marks)

5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.

1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000$$ (8)
2. Find the speed of the rocket when burning stops.
   (6) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4c585ec7-7b3e-4ff8-b7c2-83f58ad82ae9-4_316_929_391_573}
   \end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.

4. A rocket-driven car propels itself forwards in a straight line on a horizontal track by ejecting burnt fuel backwards at a constant rate \(\lambda \mathrm { kg } \mathrm { s } ^ { - 1 }\) and at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the car. At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the total resistance to the motion of the car has magnitude \(k v \mathrm {~N}\), where \(k\) is a positive constant. When \(t = 0\) the total mass of the car, including fuel, is \(M \mathrm {~kg}\). Assuming that at time \(t\) seconds some fuel remains in the car,

1. show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }$$
2. find the speed of the car at time \(t\) seconds, given that it starts from rest when \(t = 0\) and that \(\lambda = k = 10\).

4. At time \(t = 0\) a rocket is launched from rest vertically upwards. The rocket propels itself upwards by expelling burnt fuel vertically downwards with constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket. The initial mass of the rocket is \(M _ { 0 } \mathrm {~kg}\). At time \(t\) seconds, where \(t < 2\), its mass is \(M _ { 0 } \left( 1 - \frac { 1 } { 2 } t \right) \mathrm { kg }\), and it is moving upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { U } { ( 2 - t ) } - 9.8 .$$
2. Hence show that \(U > 19.6\).
3. Find, in terms of \(U\), the speed of the rocket one second after its launch.

A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed \(c\) relative to the spaceship. When the speed of the spaceship is \(v\), its mass is \(m\).

1. Show that, while the spaceship is ejecting fuel, $$\frac { \mathrm { d } v } { \mathrm {~d} m } = - \frac { c } { m } .$$ The initial mass of the spaceship is \(m _ { 0 }\) and at time \(t\) the mass of the spaceship is given by \(m = m _ { 0 } ( 1 - k t )\), where \(k\) is a positive constant.
2. Find the acceleration of the spaceship at time \(t\).

A raindrop falls vertically under gravity through a cloud. In a model of the motion the raindrop is assumed to be spherical at all times and the cloud is assumed to consist of stationary water particles. At time \(t = 0\), the raindrop is at rest and has radius \(a\). As the raindrop falls, water particles from the cloud condense onto it and the radius of the raindrop is assumed to increase at a constant rate \(\lambda\). A time \(t\) the speed of the raindrop is \(v\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 3 \lambda v } { ( \lambda t + a ) } = g$$
2. Find the speed of the raindrop when its radius is \(3 a\).
   

5 A car of mass 1600 kg is travelling uphill along a straight road inclined at \(4.7 ^ { \circ }\) to the horizontal. The power developed by the car is constant and equal to 120 kW . The car is towing a caravan and together they have a maximum speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) uphill. In this question you may model any resistances to motion as negligible.

1. Determine the mass of the caravan. The caravan is now detached from the car. Continuing up the same road, the car passes a point A at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car later passes through a point \(B\) on the same road such that \(A B = 80 \mathrm {~m}\) and the car takes 3.54 seconds to travel from A to B . The power developed by the car while travelling from A to B is constant and equal to 80 kW .
2. Determine the speed of the car at B .
3. State one possible refinement to the model used in parts (a) and (b).

2 A particle P of mass \(m\) travels in a straight line on a smooth horizontal surface.
At time \(t , \mathrm { P }\) is a distance \(x\) from a fixed point O and is moving with speed \(v\) away from O . A horizontal force of magnitude \(3 m t\) acts on P , in a direction away from O .

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 3 t\).
2. Verify that the general solution of this differential equation is \(x = \frac { 1 } { 2 } t ^ { 3 } + A t + k\), where \(A\) and \(k\) are constants.
3. Given that \(x = 6\) and \(v = 12\) when \(t = 1\), find the values of \(A\) and \(k\).

3 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors and \(c\) is a positive real number.
The resultant of two forces \(c \mathbf { i N }\) and \(- \mathbf { i } + 2 \sqrt { c } \mathbf { j N }\) is denoted by \(R \mathrm {~N}\).

1. Show that the magnitude of \(R\) is \(c + 1\). A car of mass 900 kg travels along a straight horizontal road with constant resistance to motion of magnitude \(( c + 1 ) \mathrm { N }\). The car passes through point A on the road with speed \(6 \mathrm {~ms} ^ { - 1 }\), and 8 seconds later passes through a point B on the same road. The power developed by the car while travelling from A to B is zero. Furthermore, while travelling between A and B, the car's direction of motion is unchanged.
2. Determine the range of possible values of \(c\). The car later passes through a point C on the road. While travelling between B and C the power developed by the car is modelled as constant and equal to 18 kW . The car passes through C with speed \(5 \mathrm {~ms} ^ { - 1 }\) and acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\).
3. Determine the value of \(c\).
4. Suggest how one of the modelling assumptions made in this question could be improved.

4. A car of mass 1200 kg has an engine that is capable of producing a maximum power of 80 kW . When in motion, the car experiences a constant resistive force of 2000 N .

1. Calculate the maximum possible speed of the car when travelling on a straight horizontal road.
2. The car travels up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). If the car's engine is working at \(80 \%\) capacity, calculate the acceleration of the car at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).
3. Explain why the assumption of a constant resistive force may be unrealistic.

A large aeroplane, of mass 360 tonnes, starts from rest at the beginning of a straight horizontal runway. The aeroplane produces a constant thrust of 980 kN and experiences a variable resistance to motion of magnitude \(\left( 80 + 0 \cdot 1 v ^ { 2 } \right) \mathrm { kN }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the aeroplane after it has travelled \(x\) metres.

1. 1. Find the maximum speed that the aeroplane can attain.
   2. Show that \(v\) satisfies the differential equation $$3600 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 9000 - v ^ { 2 } .$$
   (b) Find an expression for \(v ^ { 2 }\) in terms of \(x\).
2. Given that the aeroplane must achieve a speed of at least \(85 \mathrm {~ms} ^ { - 1 }\) to take off, determine the minimum length of the runway.
3. Explain why, according to this model, the aeroplane will not reach the speed found in (a)(i).

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 ball of mass 0.4 kg is thrown vertically upwards from a point \(O\) with initial speed \(17 \mathrm {~ms} ^ { - 1 }\). When the ball is at a height of \(x \mathrm {~m}\) above \(O\) and its speed is \(v \mathrm {~ms} ^ { - 1 }\), the air resistance acting on the ball has magnitude \(0.01 v ^ { 2 } \mathrm {~N}\).

1. Show that, as the ball is ascending, \(v\) satisfies the differential equation $$40 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 392 + v ^ { 2 } \right)$$
2. Find an expression for \(v\) in terms of \(x\).
3. Calculate, correct to two decimal places, the greatest height of the ball.
4. State, with a reason, whether the speed of the ball when it returns to \(O\) is greater than \(17 \mathrm {~ms} ^ { - 1 }\), less than \(17 \mathrm {~ms} ^ { - 1 }\) or equal to \(17 \mathrm {~ms} ^ { - 1 }\).
   

3. A body, of mass 9 kg , is projected along a straight horizontal track with an initial speed of \(20 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) the body experiences a resistance of magnitude \(( 0.2 + 0.03 v ) \mathrm { N }\) where \(v \mathrm {~ms} ^ { - 1 }\) is its speed.

1. Show that \(v\) satisfies the differential equation $$900 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( 20 + 3 v )$$
2. Find an expression for \(t\) in terms of \(v\).
3. Calculate, to the nearest second, the time taken for the body to come to rest.

1. A particle, \(P\), moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the direction of \(x\) increasing.

When \(t = 0\) the particle is at rest at the origin \(O\).
Given that \(a = \frac { 5 } { 2 } ( 5 - v )\)

1. show that \(v = 5 \left( 1 - \mathrm { e } ^ { - 2.5 t } \right)\)
2. state the limiting value of \(v\) as \(t\) increases. At the instant when \(v = 2.5\), the particle is \(d\) metres from \(O\).
3. Show that \(d = 2 \ln 2 - 1\)

1. A car moves in a straight line along a horizontal road. The car is modelled as a particle. At time \(t\) seconds, where \(t \geqslant 0\), the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\)

At the instant when \(t = 0\), the car passes through the point \(A\) with speed \(2 \mathrm {~ms} ^ { - 1 }\) The acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the car is modelled by $$a = \frac { 4 } { 2 + v }$$ in the direction of motion of the car.

1. Use algebraic integration to show that \(v = \sqrt { 8 t + 16 } - 2\) At the instant when the car passes through the point \(B\), the speed of the car is \(4 \mathrm {~ms} ^ { - 1 }\)
2. Use algebraic integration to find the distance \(A B\).