N2017 P1 Q11

Sir Isaac Newton was a famous scientist renowned for his work on the laws of motion. One law states that, for an object falling vertically in a vacuum, the rate of change of velocity, \(v\) ms\(^{-1}\), with respect to time, \(t\) seconds, is a constant, \(c\).

1. 1. Write down a differential equation relating \(v\), \(t\) and \(c\).[1]
   2. Initially the velocity of the object is \(4\) ms\(^{-1}\) and, after a further \(2.5\) s, the velocity of the object is \(29\) ms\(^{-1}\). Find \(v\) in terms of \(t\) and state the value of \(c\).[3]

For an object falling vertically through the atmosphere, the rate of change of velocity is less than that for an object falling in a vacuum. The new rate of change of \(v\) is modelled as the difference between the value of \(c\) found in part (i)(b) and an amount proportional to the velocity \(v\), with a constant of proportionality \(k\).

2. Given that in this case the initial velocity is zero, find \(v\) in terms of \(t\) and \(k\).[5]

For an object falling through the atmosphere, the ‘terminal velocity’ is the value approached by the velocity after a long time.

3. A falling object has initial velocity zero and terminal velocity \(40\) ms\(^{-1}\). Find how long it takes the object to reach \(90\%\) of its terminal velocity.[4]