2022 DHS P1 Q10

An object is moving from rest in a gas chamber and \(t\) seconds later, its velocity \(v\) metres per second satisfies the differential equation

\(\frac{\mathrm{d}v}{\mathrm{d}t}=5-0.2{{v}^{2}}\)

It is given that \(\frac{\mathrm{d}v}{\mathrm{d}t}>0\) and \(v\ge 0\) for all values of \(t\).

1. Find \(t\) in terms of \(v\), simplifying your answer.[5]
2. Describe how the velocity of the object varies with time.[3]
3. The displacement of the object, in metres, at any time \(m\) seconds is given by \(\displaystyle\displaystyle \displaystyle\int_{0}^{m}{v\mathrm{ d}t}\). By evaluating this integral, show that the displacement of the object is \(5\ln \left( \frac{{{e}^{m}}+{{e}^{-m}}}{2} \right)\).[4]