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I've been working on the following problem for quite some time, with no success:

A commuter train travels between two downtown stations. Because the stations are only 1.15 km apart, the train never reaches its maximum possible cruising speed. The engineer minimizes the time t between the two stations by accelerating at a rate a1 = 0.105 m/s^2 for a time t1 and then by braking with acceleration a2 = -0.585 m/s^2 for a time t2. Calculate the minimum time of travel t.

I've approached the problem in a few different ways; i've been attempting to state the supplied information in such a way that I can treat it as a max/min optimization problem and differentiate from there. I've been setting up the information using the equation:

xfinal = xinitial + velinitial(t) + 1/2(a)t^2

for example where for the first interval t1:

xfinal = 1150m

xinitial = 0

vinitial = 0

accel = 0.105m/s^2

time = the unknown

I'm running into the conceptual problem of how to deal with the two intervals. Where the time as I've laid it out is referring to just one of the two intervals, where I need to be dealing with both, i.e. the total time. I hope someone can give me an idea of how I'm misconceiving this problem. I feel like there needs to be another equation relating the two time intervals, but I can't figure out how I need to state the variables and how to integrate both equations.

I hope I've shown enough here to incline someone to give me some guidance, because I've really run up against a brick wall on this one, and I've spent a lot of time with it.

Thank you for your time!

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# Need help with one dimensional kinematics problem

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