Resistance as function - finding acceleration or speed

A question is this type if and only if resistance is given as a function of speed (kv, kv², or other form) with k already known or given, and requires finding instantaneous acceleration, maximum speed, or speed at a given instant.

1 questions · Challenging +1.2

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OCR M3 2009 June Q4
11 marks Challenging +1.2
A motor-cycle, whose mass including the rider is \(120\) kg, is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \text{ m s}^{-1}\) and when it has travelled a distance of \(x\) m beyond \(A\) its speed is \(v \text{ m s}^{-1}\). The engine develops a constant power of \(8\) kW and resistances are modelled by a force of \(0.25v^2\) N opposing the motion.
  1. Show that \(\frac{480v^2}{v^3 - 32000} \frac{dv}{dx} = -1\). [5]
  2. Find the speed of the motor-cycle when it has travelled \(500\) m beyond \(A\). [6]