True Tears Question thread

[TrueTears]

1. Two railway stations are 2km apart. A train starts from station with an acceleration of x m/s^2 for a certain time and then retards at 2x m/s^2 until it reaches the next station. It travels for a total time of 2 minutes. Find the value of x and the greatest speed attained.

2. A train passes a station A at 30 km/h, maintains this speed for 7km and is then uniformly retarded to stop at B, 8km from A. A second train starts from A at the instant the first train passes it. It is uniformly accelerated for part of the way and uniformly retarded at the same rate for the rest of the way to stop at B at the same time that the first train stops. Find the greatest speed of the second train.

Thanks.

[Collin Li]

Question 1

Draw a velocity-time graph. The area underneath must add up to 2000, and it will be some kind of triangle, with two of the vertices at and , where is measured in seconds. The other vertex will lie somewhere in from and will be , as the train first accelerates. Let's call this vertex:

From the graph, you should be able to see that there is a maximum velocity attained at that last vertex I described.

The velocity-time graph is made up of two linear sections - an accelerating part, and a decelerating part. The acceleration is represented by the gradients of these linear sections. Since the deceleration (retardation) is double the acceleration ( vs ), then therefore the time taken for the first section (flatter gradient) is twice as much as the second section (steeper gradient).

So, therefore, (two-thirds of the way), since the first part of movement takes 2 times longer than the second part (i.e.: a 2:1 ratio - ).

Also, we know the triangles have a total area of



Therefore, using and , then: