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June 2017 p43 q3
3813
A train travels between two stations, A and B. The train starts from rest at A and accelerates at a constant rate for T seconds until it reaches a speed of 25 m s-1. It then travels at this constant speed before decelerating at a constant rate, coming to rest at B. The magnitude of the train’s deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s.
(i) Sketch the velocity-time graph for the train’s journey from A to B.
(ii) Find an expression, in terms of T, for the length of time for which the train is travelling with constant speed.
(iii) The distance from A to B is 3300 m. Find how far the train travels while it is decelerating.
Solution
(i) The velocity-time graph is a trapezium. The train accelerates to 25 m/s, travels at constant speed, then decelerates to rest. The right-hand slope is steeper than the left-hand slope because the deceleration is twice the acceleration.
(ii) Let the acceleration be a. Then, the deceleration is 2a. The time to decelerate from 25 m/s to 0 is T/2. The total time is 180 s, so the time at constant speed is:
\(180 - T - \frac{T}{2} = 180 - 1.5T\)
(iii) The total distance is 3300 m. Using the area under the velocity-time graph, the distance is: