Train journey with a speed limit

A train leaves a station from rest and accelerates uniformly at \(0.5\) m/s\(^2\) to a cruising speed of \(V\) m/s. It travels at this speed and then decelerates uniformly at \(1\) m/s\(^2\), coming to rest at the next station. Model the train as moving along a straight track.

1. Using a speed–time graph, show that if the whole journey takes \(150\) s, the distance travelled is \((150V-1.5V^2)\) m.[3]
2. The stations are \(2\) km apart and the journey takes \(150\) s. Find \(V\), explaining why one solution of your equation must be rejected.[3]
3. The speed limit on the track is \(20\) m/s. With the same acceleration and deceleration, find the shortest possible time for the \(2\) km journey, justifying that it is the shortest.[3]