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

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 a magnitude which is inversely proportional to \(x^2\). 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 acting on \(P\) is \(\frac{mgR^2}{x^2}\) [2]

The particle was fired with initial speed \(U\) and the greatest height above the surface of the Earth reached by \(P\) is \(\frac{R}{20}\). Given that air resistance can be ignored,

1. find \(U\) in terms of \(g\) and \(R\). [7]

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

1. Prove that the magnitude of the gravitational force on \(P\) is \(\frac{mgR^2}{x^2}\). [3]

A particle is fired vertically upwards from the surface of the Earth with initial speed \(3U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).

1. Find \(U\) in terms of \(g\) and \(R\). [7]

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

1. Show that at time \(t\) seconds the velocity of \(P\) is \(\left(\frac{3}{t+1} - 1\right)\) m s\(^{-1}\). [5]
2. Find, to 3 significant figures, the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest. [7]

A particle \(P\) of mass 0.2 kg moves away from the origin along the positive \(x\)-axis. It moves under the action of a force directed away from the origin \(O\), of magnitude \(\frac{5}{x+1}\) N, where \(OP = x\) metres. Given that the speed of \(P\) is 5 m s\(^{-1}\) when \(x = 0\), find the value of \(x\), to 3 significant figures, when the speed of \(P\) is 15 m s\(^{-1}\). [8]

At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed 18 m s\(^{-1}\) along the \(x\)-axis, in the positive \(x\)-direction. At time \(t\) seconds (\(t > 0\)) the acceleration of \(P\) has magnitude \(\frac{3}{\sqrt{(t + 4)}}\) m s\(^{-2}\) and is directed towards \(O\).

1. Show that, at time \(t\) seconds, the velocity of \(P\) is \([30 - 6\sqrt{(t + 4)}]\) m s\(^{-1}\). [5]
2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest. [7]

A particle \(P\) moves on the positive \(x\)-axis. When the distance of \(P\) from the origin \(O\) is \(x\) metres, the acceleration of \(P\) is \((7 - 2x)\) m s\(^{-2}\), measured in the positive \(x\)-direction. When \(t = 0\), \(P\) is at \(O\) and is moving in the positive \(x\)-direction with speed 6 m s\(^{-1}\). Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest. [6]

A particle \(P\) moves along the x-axis in the positive direction. At time \(t\) seconds, the velocity of \(P\) is \(v\) m s\(^{-1}\) and its acceleration is \(\frac{1}{5}e^{-2t}\) m s\(^{-2}\). When \(t = 0\) the speed of \(P\) is 10 m s\(^{-1}\).

1. Express \(v\) in terms of \(t\). [4]
2. Find, to 3 significant figures, the speed of \(P\) when \(t = 3\). [2]
3. Find the limiting value of \(v\). [1]

A projectile \(P\) is fired vertically upwards from a point on the earth's surface. When \(P\) is at a distance \(x\) from the centre of the earth its speed is \(v\). Its acceleration is directed towards the centre of the earth and has magnitude \(\frac{k}{x^2}\), where \(k\) is a constant. The earth may be assumed to be a sphere of radius \(R\).

1. Show that the motion of \(P\) may be modelled by the differential equation $$v \frac{dv}{dx} = -\frac{gR^2}{x^2}.$$ [4]

The initial speed of \(P\) is \(U\), where \(U^2 < 2gR\). The greatest distance of \(P\) from the centre of the earth is \(X\).

1. Find \(X\) in terms of \(U\), \(R\) and \(g\). [6]

A particle \(P\) of mass 2.5 kg moves along the positive \(x\)-axis. It moves away from a fixed origin \(O\), under the action of a force directed away from \(O\). When \(OP = x\) metres the magnitude of the force is \(2e^{-0.1x}\) newtons and the speed of \(P\) is \(v\) m s\(^{-1}\). When \(x = 0\), \(v = 2\). Find

1. \(v^2\) in terms of \(x\), [6]
2. the value of \(x\) when \(v = 4\). [3]
3. Give a reason why the speed of \(P\) does not exceed \(\sqrt{20}\) m s\(^{-1}\). [1]

A light elastic string \(AB\) of natural length 1.5 m has modulus of elasticity 20 N. The end \(A\) is fixed to a point on a smooth horizontal table. A small ball \(S\) of mass 0.2 kg is attached to the end \(B\). Initially \(S\) is at rest on the table with \(AB = 1.5\) m. The ball \(S\) is then projected horizontally directly away from \(A\) with a speed of 5 m s\(^{-1}\). By modelling \(S\) as a particle,

1. find the speed of \(S\) when \(AS = 2\) m. [5]

When the speed of \(S\) is 1.5 m s\(^{-1}\), the string breaks.

1. Find the tension in the string immediately before the string breaks. [5]

A toy car of mass \(0.2\) kg is travelling in a straight line on a horizontal floor. The car is modelled as a particle. At time \(t = 0\) the car passes through a fixed point \(O\). After \(t\) seconds the speed of the car is \(v \text{ m s}^{-1}\) and the car is at a point \(P\) with \(OP = x\) metres. The resultant force on the car is modelled as \(\frac{1}{5}x(4 - 3x)\) N in the direction \(OP\). The car comes to instantaneous rest when \(x = 6\). Find

1. an expression for \(v^2\) in terms of \(x\), [7]
2. the initial speed of the car. [2]

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

1. Show that \(k = mgR^2\). [2]

Given that \(S\) starts from rest when its distance from the centre of the earth is \(2R\), and that air resistance can be ignored,

1. find the speed of \(S\) as it crashes into the surface of the earth. [7]

A cyclist and her bicycle have a combined mass of \(100\) kg. She is working at a constant rate of \(80\) W and is moving in a straight line on a horizontal road. The resistance to motion is proportional to the square of her speed. Her initial speed is \(4\) m s\(^{-1}\) and her maximum possible speed under these conditions is \(20\) m s\(^{-1}\). When she is at a distance \(x\) m from a fixed point \(O\) on the road, she is moving with speed \(v\) m s\(^{-1}\) away from \(O\).

1. Show that $$v \frac{dv}{dx} = \frac{8000 - v^3}{10000v}.$$ [5]
2. Find the distance she travels as her speed increases from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [5]
3. Use the trapezium rule, with 2 intervals, to estimate how long it takes for her speed to increase from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [4]

A particle \(P\) of mass \(0.25\) kg is moving along the positive \(x\)-axis under the action of a single force. At time \(t\) seconds \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v\) m s\(^{-1}\) where \(\frac{\mathrm{d}v}{\mathrm{d}x} = 3\). It is given that \(x = 2\) and \(v = 3\) when \(t = 0\)

1. Find the magnitude of the force acting on \(P\) when \(x = 5\) [4]
2. Find the value of \(t\) when \(x = 5\) [4]

A puck, of mass \(m\) kg, is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \text{ m s}^{-1}\). As the puck moves, it experiences an air resistance force of magnitude \(0.3mv^3\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \text{ m s}^{-1}\). Model the puck as a particle.

1. Show that $$v = (4 - 0.2t)^{\frac{3}{2}}$$ [6 marks]
2. Find the value of \(t\) when the puck comes to rest. [2 marks]
3. Find the distance travelled by the puck as its speed decreases from \(8 \text{ m s}^{-1}\) to zero. [5 marks]

A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda mv\), where \(\lambda\) is a constant.

1. Show that $$\frac{\text{d}v}{\text{d}t} = g - \lambda v$$ [2 marks]
2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\). [6 marks]

A stone, of mass 0.9 kg, is projected vertically upwards with speed 10 ms\(^{-1}\) in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m. Find the magnitude of the non-gravitational resisting force acting on the stone. [5 marks]

A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x\) m from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac{8}{x}\) N. When \(x = 2\), the speed of \(P\) is 4 ms\(^{-1}\). Find the speed of \(P\) when it is 0.5 m from \(O\). [8 marks]

A particle of mass \(m\) kg moves in a horizontal straight line. Its initial speed is \(u\) ms\(^{-1}\) and the only force acting on it is a variable resistance of magnitude \(mkv\) N, where \(v\) ms\(^{-1}\) is the speed of the particle after \(t\) seconds and \(k\) is a constant. Show that \(v = ue^{-kt}\). [7 marks]

The acceleration \(a\) ms\(^{-2}\) of a particle \(P\) moving in a straight line away from a fixed point \(O\) is given by \(a = \frac{k}{1+t}\), where \(t\) is the time that has elapsed since \(P\) left \(O\), and \(k\) is a constant.

1. By solving a suitable differential equation, find an expression for the velocity \(v\) ms\(^{-1}\) of \(P\) in terms of \(t\), \(k\) and another constant \(c\). [3 marks]

Given that \(v = 0\) when \(t = 0\) and that \(v = 4\) when \(t = 2\),

1. show that \(v \ln 3 = 4 \ln (1 + t)\). [3 marks]
2. Calculate the time when \(P\) has a speed of 8 ms\(^{-1}\). [3 marks]

A particle of mass \(m\) kg, at a distance \(x\) m from the centre of the Earth, experiences a force of magnitude \(\frac{km}{x^2}\) N towards the centre of the Earth, where \(k\) is a constant. Given that the radius of the Earth is \(6.37 \times 10^6\) m, and that a 3 kg mass experiences a force of 30 N at the surface of the Earth,

1. calculate the value of \(k\), stating the units of your answer. [3 marks]

The 3 kg mass falls from rest at a distance \(x = 12.74 \times 10^6\) m from the centre of the Earth. Ignoring air resistance,

1. show that it reaches the surface of the Earth with speed \(7.98 \times 10^3\) ms\(^{-1}\). [7 marks]

In a simplified model, the particle is assumed to fall with a constant acceleration 10 ms\(^{-2}\). According to this model it attains the same speed as in (b), \(7.98 \times 10^3\) ms\(^{-1}\), at a distance \((12.74 - d) \times 10^6\) m from the centre of the Earth.

1. Find the value of \(d\). [3 marks]

A small sphere \(S\), of mass \(m\) kg is released from rest at the surface of a liquid in a right circular cylinder whose axis is vertical. When \(S\) is moving downwards with speed \(v\) ms\(^{-1}\), the viscous resistive force acting upwards on it has magnitude \(v^2\) N.

1. Write down a differential equation for the motion of \(S\), clearly defining any symbol(s) that you introduce. [4 marks]
2. Find, in terms of \(m\), the distance \(S\) has fallen when its speed is \(\sqrt{\frac{mg}{2}}\) ms\(^{-1}\). [9 marks]

The diagram shows two identical particles, each of mass \(m\) kg, connected by a thin, light inextensible string. \(P\) slides on the surface of a smooth right circular cylinder fixed with its axis, through \(O\), horizontal. \(Q\) moves vertically. \(OP\) makes an angle \(\theta\) radians with the horizontal. \includegraphics{figure_6} The system is released from rest in the position where \(\theta = 0\).

1. Show that the vertical distance moved by \(Q\) is \(\frac{\theta}{\sin \theta}\) times the vertical distance moved by \(P\). [4 marks]
2. In the position where \(\theta = \frac{\pi}{6}\), prove that the reaction of the cylinder on \(P\) has magnitude \(\left(1-\frac{\pi}{6}\right)mg\) N. [9 marks]

A particle \(P\) of mass \(m\) kg moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude \(m f(x)\) N, where \(f(x)\) is a function of the height \(x\) m of \(P\) above the ground. When \(P\) is at this height, its upward speed \(v\) ms\(^{-1}\) is given by $$v^2 = 2e^{-2gx} - 1.$$

1. Write down a differential equation for the motion of \(P\) and hence determine \(f(x)\) in terms of \(g\) and \(x\). [5 marks]
2. Show that the greatest height reached by \(P\) above the ground is \(\frac{1}{2g} \ln 2\) m. [2 marks]

Given that the work, in J, done by \(P\) against the resisting force as it moves from ground level to a point \(H\) m above the ground is equal to \(\int_0^H m f(x) dx\),

1. show that the total work done by \(P\) against the resistance during its upward motion is \(\frac{1}{2}m(1 - \ln 2)\) J. [3 marks]

A particle \(P\) of mass \(m\) kg moves along a straight line under the action of a force of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant, directed towards a fixed point \(O\) on the line, where \(OP = x\) m. \(P\) starts from rest at \(A\), at a distance \(a\) m from \(O\). When \(OP = x\) m, the speed of \(P\) is \(v\) ms\(^{-1}\).

1. Show that \(v = \sqrt{\frac{2k(a-x)}{ax}}\). [6 marks]

\(B\) is the point half-way between \(O\) and \(A\). When \(k = \frac{1}{2}\) and \(a = 1\), the time taken by \(P\) to travel from \(A\) to \(B\) is \(T\) seconds Assuming the result that, for \(0 \leq x \leq 1\), \(\int \sqrt{\frac{x}{1-x}} dx = \arcsin(\sqrt{x}) - \sqrt{x(1-x^2)} + \text{constant}\),

1. find the value of \(T\). [5 marks]