Please help with this question

Question

A train X runs on a straight horizontal track that connects stations A and C. Station B lies between A and C. Figure 2 shows the graph of the speed, v m s−1, of train X against the time, t minutes, after noon.

Train X leaves A at noon and accelerates uniformly from rest until t = 1, when it is moving with speed 16 m s−1. 

Train X then continues to move along the track at constant speed for 15 minutes, before decelerating uniformly and coming to rest at B at time t = 17.

Train X leaves B at time t = 21and accelerates uniformly for one minute, reaching a speed of 16 m s−1. 

Train X then moves along the track at a constant speed of 16 m s−1 for 10 minutes, before decelerating uniformly and coming to rest at C at time t = 33.

(a)   Find the distance of C from A, stating the units of your answer. 

(3)

A second train, Y, leaves A at T minutes after noon and moves in the same direction as train X on a parallel straight horizontal track.

Train Y accelerates uniformly from rest for 2 minutes, reaching a speed of 24 m s−1.

Train Y then moves along the track at a constant speed of 24 m s−1 and passes C at this speed.

Train Y passes C at the instant train X stops at C.

(b)   Find the value of T. 

(5)

(c)   State one assumption made in your working that could affect the accuracy of your answer to part (b).

(1)

Answer 1

Turn all the information into math, as follows.

"from rest" x(0) = 0.

"accelerates uniformly" x'' (t) = a > 0 for 0 < t < 1.

x'(1) = 16.

"at constant speed" x''(t) = 16 for 1 < t < 16.

"decelerating uniformly" x'' (t) = –b < 0 for 16 < t < 17.

"coming to rest" x' (t) = 0 for 17 < t < 21.

At this point we can quickly obtain x(t) = at²/2 + ct for 0 < t < 1, a + c = 16.

Also x(t) = a/2 + c + 16(t – 1) for 1 < t < 16, and

x(t) = a/2 + c + 16•15 – b(t – 16)²/2 + d (t – 16) for 16 < t < 17, –b + d = 0.

Check that I made no mistakes, but the idea going forward is to keep chaining the information together like this (keeping x(t) a continuous function) so there are enough equations to determine whatever is unknown.