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

A particle of mass \(m\) moves in a straight line. At time \(t\), its displacement from a fixed point on the line is \(s\) and its velocity is \(v\). The particle experiences a retarding force of magnitude \(mkv^2\), where \(k\) is a positive constant. Find the relationship between \(v\) and \(t\). [7]

\(O\), \(A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass \(0.6\) kg moves along the line. At time \(t\) s the particle has displacement \(x\) m from \(O\) and speed \(v\) m s\(^{-1}\). The only horizontal force acting on \(P\) has magnitude \(0.4v^{\frac{1}{2}}\) N and acts in the direction \(OA\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).

1. Show that \(3v^{\frac{1}{2}}\frac{dv}{dx} = 2\). [2]
2. Express \(v\) in terms of \(x\). [4]
3. Given that \(AB = 7\) m, find the value of \(t\) when \(P\) passes through \(B\). [3]

A particle \(P\) of mass \(0.3\text{ kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\text{ m s}^{-1}\). A force of magnitude \(2x\text{ N}\) acts on \(P\) in the direction \(PO\), where \(x\text{ m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v\frac{\text{d}v}{\text{d}x} = kx\) and state the value of the constant \(k\). [2]
2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]

A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\,\text{m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\,\text{m s}^{-1}\) and the acceleration of \(P\) is given by \(e^{-0.5t}\,\text{m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]

A particle \(P\) of mass \(0.3\,\text{kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\,\text{m s}^{-1}\). A force of magnitude \(2x\,\text{N}\) acts on \(P\) in the direction \(PO\), where \(x\,\text{m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v\frac{dv}{dx} = kx\) and state the value of the constant \(k\). [2]
2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [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 \(v\) m s\(^{-1}\) when its displacement is \(x\) m from \(O\). The force acting on \(B\) has magnitude \((2 + 0.3x^2)\) N and is directed horizontally away from \(O\).

1. Show that \(v\frac{dv}{dx} = 1.2x^2 + 8\). [2]
2. Find the velocity of \(B\) when \(x = 1.5\). [3]

An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v\frac{dv}{dx} = 1.2x^2 + 6 - 3x.$$

1. Find the magnitude of this extra force and state the direction in which it acts. [2]

\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).

1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3]
2. The maximum value of \(v\) is \(4.5\). Find the initial value of \(v\). [5]

\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).

1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3] The maximum value of \(v\) is \(4.5\).
2. Find the initial value of \(v\). [5]

A particle \(P\) is moving along a straight line with acceleration \(3ku - kv\) where \(v\) is its velocity at time \(t\), \(u\) is its initial velocity and \(k\) is a constant. The velocity and acceleration of \(P\) are both in the direction of increasing displacement from the initial position.

1. Find the time taken for \(P\) to achieve a velocity of \(2u\). [3]
2. Find an expression for the displacement of \(P\) from its initial position when its velocity is \(2u\). [5]

A particle \(Q\) of mass \(m\) kg falls from rest under gravity. The motion of \(Q\) is resisted by a force of magnitude \(mkv\) N, where \(v\) ms\(^{-1}\) is the speed of \(Q\) at time \(t\) s and \(k\) is a positive constant. Find an expression for \(v\) in terms of \(g\), \(k\) and \(t\). [6]

A particle \(P\) of mass 1 kg is moving along a straight line against a resistive force of magnitude \(\frac{10\sqrt{v}}{(t+1)^2}\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) at time \(t\)s. When \(t = 0\), \(v = 25\). Find an expression for \(v\) in terms of \(t\). [5]

A particle \(P\) of mass \(m\) kg is projected vertically upwards from a point \(O\), with speed \(20\) m s\(^{-1}\), and moves under gravity. There is a resistive force of magnitude \(2mv\) N, where \(v\) m s\(^{-1}\) is the speed of \(P\) at time \(t\) s after projection.

1. Find an expression for \(v\) in terms of \(t\), while \(P\) is moving upwards. [6]

The displacement of \(P\) from \(O\) is \(x\) m at time \(t\) s.

1. Find an expression for \(x\) in terms of \(t\), while \(P\) is moving upwards. [2]
2. Find, correct to 3 significant figures, the greatest height above \(O\) reached by \(P\). [2]

A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \text{ m s}^{-1}\). At time \(t\) s after passing through \(O\), the acceleration of \(P\) is \(-\frac{4000}{(5t + 4)^3} \text{ m s}^{-2}\) in the direction \(PO\). The displacement of \(P\) from \(O\) at time \(t\) is \(x\) m. Find an expression for \(x\) in terms of \(t\). [5]

A particle \(P\) moving in a straight line has displacement \(x\)m from a fixed point \(O\) on the line and velocity \(v\)m s\(^{-1}\) at time \(t\)s. The acceleration of \(P\), in m s\(^{-2}\), is given by \(6\sqrt{v + 9}\). When \(t = 0\), \(x = 2\) and \(v = 72\).

1. Find an expression for \(v\) in terms of \(x\). [4]
2. Find an expression for \(x\) in terms of \(t\). [5]

A particle of mass \(m\) kg falls vertically under gravity, from rest. At time \(t\) s, \(P\) has fallen \(x\) m and has velocity \(v\) m s\(^{-1}\). The only forces acting on \(P\) are its weight and a resistance of magnitude \(kmgv\) N, where \(k\) is a constant.

1. Find an expression for \(v\) in terms of \(t\), \(g\) and \(k\). [5]
2. Given that \(k = 0.05\), find, in metres, how far \(P\) has fallen when its speed is \(12\) m s\(^{-1}\). [5]

A particle \(P\) of mass \(2\) kg moving on a horizontal straight line has displacement \(x\) m from a fixed point \(O\) on the line and velocity \(v\) m s\(^{-1}\) at time \(t\) s. The only horizontal force acting on \(P\) has magnitude \(\frac{1}{10}(2v - 1)^2e^{-t}\) N and acts towards \(O\). When \(t = 0\), \(x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass \(2\) kg moves along a horizontal straight line. The point \(O\) is a fixed point on this line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and the displacement of \(P\) from \(O\) is \(x\) m. A force of magnitude \(\left(8x - \frac{128}{x^3}\right)\) N acts on \(P\) in the direction \(OP\). When \(t = 0\), \(x = 8\) and \(v = -15\).

1. Show that \(v = -\frac{2}{3}(x^2 - 4)\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass \(5\) kg moves along a horizontal straight line. At time \(t\) s, the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The forces acting on \(P\) are a force of magnitude \(\frac{500}{v}\) N in the direction \(OP\) and a resistive force of magnitude \(\frac{1}{2}v^2\) N. When \(t = 0\), \(x = 0\) and \(v = 5\).

1. Find an expression for \(v\) in terms of \(x\). [6]
2. State the value that the speed approaches for large values of \(x\). [1]

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

1. Find an expression for \(v\) in terms of \(t\). [6]
2. Deduce what happens to \(v\) for large values of \(t\). [1]

A particle \(P\) of mass \(0.5\) kg moves in a straight line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The only forces acting on \(P\) are a force of magnitude \(\frac{150}{(x+1)^2}\) N in the direction of increasing displacement and a resistive force of magnitude \(\frac{450}{(x+1)^3}\) N. When \(t = 0\), \(x = 0\) and \(v = 20\). Find \(v\) in terms of \(x\), giving your answer in the form \(v = \frac{Ax + B}{(x + 1)}\), where \(A\) and \(B\) are constants to be determined. [6]

A particle \(P\) of mass \(2\text{kg}\) moving on a horizontal straight line has displacement \(x\text{m}\) from a fixed point \(O\) on the line and velocity \(v\text{ms}^{-1}\) at time \(t\). The only horizontal force acting on \(P\) is a variable force \(F\text{N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass \(m\) kg is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s.

1. Find an expression for \(v\) in terms of \(t\). [6]
2. Find an expression for \(v^2\) in terms of \(x\). [5]

The displacement of \(P\) from \(O\) at time \(t\) s is \(x\) m.

A particle \(P\) of mass \(2 \text{ kg}\) moving on a horizontal straight line has displacement \(x \text{ m}\) from a fixed point \(O\) on the line and velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\). The only horizontal force acting on \(P\) is a variable force \(F \text{ N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

A particle \(P\) of mass \(mk\) falls from rest due to gravity. There is a resistance force of magnitude \(mkv^2\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) after it has fallen a distance \(x\) m and \(k\) is a positive constant.

1. By using \(v \frac{dv}{dx} = \frac{dv}{dt}\) and appropriate differential equation, show that $$v^2 = \frac{g}{k}(1 - e^{-2kx}).$$ [7] It is given that \(k = 0.01\). The speed of \(P\) when \(x = 0.2\) comes to approximately \(v\) ms\(^{-1}\).
2. 1. Find \(V\) correct to 2 decimal places. [1]
   2. Hence find how far \(P\) has fallen when its speed is \(\frac{1}{2}V\) ms\(^{-1}\). [2]