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

9 A particle of mass 2 kg is moving along the \(x\)-axis, which is horizontal, against a resistive force which is proportional to the cube of the speed of the particle at any instant. At time \(t\) seconds the particle's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement is \(x \mathrm {~m}\). When \(t = 0 , x = 0 , v = 4\) and the retardation is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that $$\frac { 1 } { v } = \frac { x + 8 } { 32 } .$$
2. Find the time taken to cover the first 8 metres.

12 A bullet of mass 0.0025 kg is fired vertically upwards from a point \(O\). At time \(t \mathrm {~s}\) after projection the speed of the bullet is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion has magnitude \(0.00001 v ^ { 2 } \mathrm {~N}\).

1. Show that, while the bullet is rising, $$250 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 2500 - v ^ { 2 }$$
2. It is given that the speed of projection is \(350 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
   1. the time taken after projection for the bullet to reach its greatest height above \(O\),
   2. the greatest height above \(O\) reached by the bullet.

10 A small body of mass \(m\) is thrown vertically upwards with initial velocity \(u\). Resistance to motion is \(k v ^ { 2 }\) per unit mass, where the velocity is \(v\) and \(k\) is a positive constant. Find, in terms of \(u , g\) and \(k\),

1. the time taken to reach the greatest height,
2. the greatest height to which the body will rise.

11 A car of mass 800 kg has a constant power output of 32 kW while travelling on a horizontal road. At time \(t \mathrm {~s}\) the car's speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistive force has magnitude \(20 v \mathrm {~N}\).

1. Show that \(v\) satisfies the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1600 - v ^ { 2 } } { 40 v }\).
2. Given that \(v = 0\) when \(t = 0\), solve this differential equation to find \(v\) in terms of \(t\). State what the solution predicts as \(t\) becomes large.

7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. If the cyclist's maximum steady speed is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

12 A particle of mass 0.6 kg is projected vertically upwards from horizontal ground, with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The upwards velocity at any instant is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the displacement is \(x \mathrm {~m}\). Air resistance is modelled by a force \(0.024 v ^ { 2 } \mathrm {~N}\) acting downwards.

1. Show that \(v\) and \(x\) satisfy the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 } .$$
2. Find the value of \(u\) if the maximum height reached is 50 m .

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

10 A cyclist and her bicycle have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.

1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
2. Show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).

3 A light spring, of natural length 0.4 m and modulus of elasticity 6.4 N , has one end \(A\) attached to the ceiling of a room. A particle of mass \(m \mathrm {~kg}\) is attached to the free end of the spring and hangs in equilibrium. The particle is displaced vertically downwards and released from rest. In the subsequent motion the particle does not reach the ceiling and air resistance may be neglected.

1. Show that the particle oscillates in simple harmonic motion.
2. Given that the period of the motion is 1.12 s , find
   1. the value of \(m\), correct to 3 significant figures,
   2. the extension of the spring when the particle has a downwards acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

5 When a car of mass 990 kg moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal straight road, the power of its engine is 8.8 kW .

1. Find the magnitude of the resistance to the motion of the car at this speed.
2. Assuming that the resistance has magnitude \(k v ^ { 2 }\) newtons when the speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of the constant \(k\). The power of the engine is now increased to 22 kW and remains constant at this value.
3. Using the model in part (ii), show that $$\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 20000 - v ^ { 3 } } { 900 v ^ { 2 } } .$$
4. Hence show that the car moves about 300 m as its speed increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A train is moving along a straight horizontal track. At time t seconds, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), its acceleration is \(a \mathrm {~ms} ^ { - 2 }\), and \(a\) is inversely proportional to V . At time \(\mathrm { t } = 1\), \(v = 5\) and \(a = 1 \cdot 8\). a) i) Write down a differential equation satisfied by V.
ii) Show that \(v ^ { 2 } = 18 t + 7\).
b) Find the time at which the magnitude of the velocity is equal to the magnitude of the acceleration. \section*{END OF PAPER}

A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \text{ m s}^{-1}\) and moves towards a fixed point \(A\) on the line. At time \(t\) s the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v \text{ m s}^{-1}\). A resistive force of magnitude \((5 - x)\) N acts on \(P\) in the direction towards \(O\).

1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2x\). [6]
2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\). [5]

A particle \(P\) of mass \(0.1\) kg moves with decreasing speed in a straight line on a smooth horizontal surface. A horizontal resisting force of magnitude \(0.2e^{-x}\) N acts on \(P\), where \(x\) m is the displacement of \(P\) from a fixed point \(O\) on the line. The velocity of \(P\) is \(v\) m s\(^{-1}\) when its displacement from \(O\) is \(x\) m.

1. Show that $$v\frac{dv}{dx} = ke^{-x},$$ where \(k\) is a constant to be found. [2]

\(P\) passes through \(O\) with velocity \(2.2\) m s\(^{-1}\).

1. Calculate the value of \(x\) at the instant when the velocity of \(P\) is \(2\) m s\(^{-1}\). [4]
2. Show that the speed of \(P\) does not fall below \(0.917\) m s\(^{-1}\), correct to \(3\) significant figures. [2]

A particle \(P\) of mass \(0.4 \text{ kg}\) is released from rest at a point \(O\) on a smooth plane inclined at \(30°\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \text{ m}\) down the plane, the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8e^{-x} \text{ N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).

1. Show that \(v \frac{dv}{dx} = 5 - 2e^{-x}\). [2]
2. Find \(v\) when \(x = 0.6\). [4]

A particle \(P\) of mass \(0.4\) kg is released from rest at a point \(O\) on a smooth plane inclined at \(30°\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x\) m down the plane, the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8e^{-5x}\) N acts on \(P\) up the plane along the line of greatest slope through \(O\).

1. Show that \(v \frac{dv}{dx} = 5 - 2e^{-x}\). [2]
2. Find \(v\) when \(x = 0.6\). [4]

A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v\,\text{m}\,\text{s}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).

1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{d}v}{\text{d}t} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]

A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v \text{ ms}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).

1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{dv}}{\text{dt}} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]

A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally along a smooth horizontal plane from a point \(O\). At time \(t \text{ s}\) after projection the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8t \text{ N}\) directed away from \(O\) acts on \(P\) and a force of magnitude \(2e^{-t} \text{ N}\) opposes the motion of \(P\).

1. Show that \(\frac{dv}{dt} = 2t - 5e^{-t}\). [2]
2. Given that \(v = 8\) when \(t = 1\), express \(v\) in terms of \(t\). [3]
3. Find the speed of projection of \(P\). [2]

A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t\) s after its release the velocity of \(P\) is 7.5 m s\(^{-1}\) downwards. A vertically downwards force of magnitude 0.6t N acts on \(P\). A vertically upwards force of magnitude \(ke^{-t}\) N, where \(k\) is a constant, also acts on \(P\).

1. Show that \(\frac{dv}{dt} = 10 - 5ke^{-t} + 3t\). [2]
2. Find the greatest value of \(k\) for which \(P\) does not initially move upwards. [3]
3. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]

\(O\) and \(A\) are fixed points on a rough horizontal surface, with \(OA = 1 \text{ m}\). A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally with speed \(U \text{ m s}^{-1}\) from \(A\) in the direction \(OA\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \text{ m}\), the velocity of \(P\) is \(v \text{ m s}^{-1}\). The coefficient of friction between the surface and \(P\) is \(0.4\). A force of magnitude \(\frac{0.8}{x} \text{ N}\) acts on \(P\) in the direction \(PO\).

1. Show that, while the particle is in motion, \(v \frac{\text{d}v}{\text{d}x} = -4 - \frac{2}{x}\). [3]

It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).

1. Find the set of possible values of \(U\). [5]

A cyclist and his bicycle have a total mass of \(81 \text{ kg}\). The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of \(135 \text{ N}\) and the motion is opposed by a resistance of magnitude \(9v \text{ N}\), where \(v \text{ m s}^{-1}\) is the cyclist's speed at time \(t \text{ s}\) after starting.

1. Show that \(\frac{9}{15-v} \frac{dv}{dt} = 1\). [2]
2. Solve this differential equation to show that \(v = 15(1-e^{-\frac{t}{9}})\). [4]
3. Find the distance travelled by the cyclist in the first \(9 \text{ s}\) of the motion. [4]

\includegraphics{figure_6} A particle \(P\) of mass \(0.2\) kg is projected with velocity \(2\) m s\(^{-1}\) upwards along a line of greatest slope on a plane inclined at \(30°\) to the horizontal (see diagram). Air resistance of magnitude \(0.5v\) N opposes the motion of \(P\), where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s after projection. The coefficient of friction between \(P\) and the plane is \(\frac{1}{2\sqrt{3}}\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).

1. Show that, while \(P\) is moving up the plane, \(\frac{dv}{dt} = -2.5(3 + v)\). [3]
2. Calculate \(T\). [4]
3. Calculate the speed of \(P\) when \(t = 2T\). [5]

A ball of mass 0.05 kg is released from rest at a height \(h\) m above the ground. At time \(t\) s after its release, the downward velocity of the ball is \(v\) m s\(^{-1}\). Air resistance opposes the motion of the ball with a force of magnitude 0.01\(v\) N.

1. Show that \(\frac{dv}{dt} = 10 - 0.2v\). Hence find \(v\) in terms of \(t\). [6]
2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\). [4]

A particle \(P\) of mass \(0.2\) kg is released from rest and falls vertically. At time \(t\) s after release \(P\) has speed \(v\) m s\(^{-1}\). A resisting force of magnitude \(0.8v\) N acts on \(P\).

1. Show that the acceleration of \(P\) is \((10 - 4v)\) m s\(^{-2}\). [2]
2. Find the value of \(v\) when \(t = 0.6\). [5]