Braking train and accelerating trolley: equal speeds, the gap between them and the distance until they meet

A train and a maintenance trolley move in the same direction along straight, parallel tracks. At the instant the back of the braking train passes a signal post, the trolley starts from rest \(150\text{ m}\) behind the post, as shown in the diagram.

At time \(t\) seconds after this instant, the velocities of the train and the trolley, in \(\text{m s}^{-1}\), are modelled by

\(v_{T}=30e^{-0.08t}\) and \(v_{M}=3t\), for \(t\geq0\).

1. Write down the speed at \(t=0\) of
   1. the train
   2. the trolley.[2]
2. Find
   1. the time at which the train and the trolley have equal speeds
   2. this common speed.[3]

Let \(s(t)\) be the distance, in metres, from the trolley to the back of the train at time \(t\), so that \(s(0)=150\).

3. Find an expression for \(s(t)\), valid until the trolley reaches the back of the train.[7]
4. Hence, find the time taken for the trolley to reach the back of the train.[2]
5. Find the distance the trolley has travelled when it reaches the back of the train.[3]