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

A particle \(P\) of mass \(2\) kg moves along the \(x\)-axis. At time \(t = 0\), \(P\) passes through the origin \(O\) with speed \(3\) m s\(^{-1}\). At time \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) m s\(^{-1}\), where \(t \geqslant 0\), \(x \geqslant 0\) and \(v \geqslant 0\). While \(P\) is in motion the only force acting on \(P\) is a resistive force \(F\) of magnitude \((v^2 + 1)\) N acting in the negative \(x\)-direction.

1. Find an expression for \(v\) in terms of \(x\). [5]
2. Determine the distance travelled by \(P\) while its speed drops from \(3\) m s\(^{-1}\) to \(2\) m s\(^{-1}\). [2]

Particle \(Q\) is identical to particle \(P\). At a different time, \(Q\) is moving along the \(x\)-axis under the influence of a single constant resistive force of magnitude \(1\) N. When \(t' = 0\), \(Q\) is at the origin and its speed is \(3\) m s\(^{-1}\).

1. By comparing the motion of \(P\) with the motion of \(Q\) explain why \(P\) must come to rest at some finite time when \(t < 6\) with \(x < 9\). [3]
2. Sketch the velocity-time graph for \(P\). You do not need to indicate any values on your sketch. [1]
3. Determine the maximum displacement of \(P\) from \(O\) during \(P\)'s motion. [2]

A particle P, of mass \(m\), moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude \(\frac{kmg}{x^2}\), where \(x\) is the distance OP. The coefficient of friction between P and the table is \(\mu\). P is initially projected in a direction directly away from O. The velocity of P is first zero at a point A which is a distance \(a\) from O.

1. Show that the velocity \(v\) of P, when P is moving away from O, satisfies the differential equation $$\frac{\mathrm{d}}{\mathrm{d}x}(v^2) + \frac{2kg}{x^2} + 2\mu g = 0.$$ [3]
2. Verify that $$v^2 = 2gk\left(\frac{1}{x} - \frac{1}{a}\right) + 2\mu g(a-x).$$ [3]
3. Find, in terms of \(k\) and \(a\), the range of values of \(\mu\) for which P remains at A. [2]

A particle P of mass 5 kg is released from rest at a point O and falls vertically. A resistance of magnitude \(0.05v^2\) N acts vertically upwards on P, where \(v \text{ m s}^{-1}\) is the velocity of P when it has fallen a distance \(x\) m.

1. Show that \(\left(\frac{100v}{980-v^2}\right)\frac{dv}{dx} = 1\). [2]
2. Verify that \(v^2 = 980(1-e^{-0.02x})\). [4]
3. Determine the work done against the resistance while P is falling from O to the point where P's acceleration is \(8.36 \text{ m s}^{-2}\). [5]

A car of mass \(900\) kg moves along a straight level road. The power developed by the car is constant and equal to \(60\) kW. The resistance to the motion of the car is constant and equal to \(1500\) N. At time \(t\) seconds the velocity of the car is denoted by \(v\) m s\(^{-1}\). Initially the car is at rest.

1. Show that \(\frac{3v\,dv}{5\,dt} = 40 - v\). [3]
2. Verify that \(t = 24\ln\left(\frac{40}{40-v}\right) - \frac{3}{5}v\). [5]

A particle, of mass 4 kg, moves in a straight line under the action of a single force \(F\) N, whose magnitude at time \(t\) seconds is given by $$F = 12\sqrt{t} - 32 \quad \text{for} \quad t \geqslant 0.$$

1. Find the acceleration of the particle when \(t = 9\). [2]
2. Given that the particle has velocity \(-1\text{ms}^{-1}\) when \(t = 4\), find an expression for the velocity of the particle at \(t\) s. [3]
3. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]

A box of mass \(2\) kg is projected along a horizontal surface with an initial velocity of \(5\) ms\(^{-1}\). The box experiences a variable resistive force of \(0.4v^2\) N, where \(v\) ms\(^{-1}\) is the velocity of the box at time \(t\) seconds.

1. Show that \(v\) satisfies the equation $$5\frac{dv}{dt} + v^2 = 0.$$ [2]
2. Find an expression for \(v\) in terms of \(t\). [4]
3. Briefly explain why this model is not particularly realistic. [1]

A particle \(P\) of mass 2 kg can only move along the straight line segment \(OA\), where \(OA\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(OA\) is 0.9 m. When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) ms\(^{-1}\). \(P\) is subject to a force of magnitude \(4e^{-2t}\) N in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2\), \(v = 0.5\) and the resultant force on \(P\) is 0 N.

1. Show that, according to the model, \(\frac{dv}{dt} + v = 2e^{-2t}\). [3]
2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\). [5]
3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\)'s speed must reach a maximum value for some \(t > 0\). [2]
4. Determine the maximum speed considered in part (c). [2]

A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v^2\).

1. Show that \(v^2 = \frac{1}{k}\left(T - Ae^{-\frac{2kx}{m}}\right)\) where \(A\) and \(k\) are constants. [5]

\(P\) starts from rest at \(O\).

1. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\). [4]
2. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases. [1]