The graph plot that you have given is 1. Desired velocity (m/sec) vs. Time (sec)  and 2. Desired distance (m) vs. Time (sec). These are the plots of a typical traction system. The area under the 1st graph gives you the desired distance. So, 2nd graph is obtained on the basis of 1st graph. Look if you have to calculate the desired distance traveled by the train at 500 sec then, you will have to find the area under the curve at same 500 sec of the 1st plot. certainly if the time goes on increasing, the area under the goes on increasing and so is the distance. So, in the 2nd plot you can observe the same.

Now coming to your question, you are asking that if you know the desired distance ,say, 37.087 Km and the time taken to travel that distance is, say, 700 sec then what will be the desired velocity (m/sec) of the train? Right  !!!!!

Now have a look at the 1st plot, at the time 700 sec , the desired velocity is zero. That means what?

That means, the 1st plot is original plot which can be different for different traction system. And, the second plot is the derived plot from the 1st one.

Now the question arises how you can have equations for curves? The answer is simple, you can apply curve fitting techniques to obtain the polynomial equations. But, this is valid for the 2nd plot only. That is, in case of 2nd plot you can obtain a fitted polynomial but still, that will not give you  the desired velocity (because of the reason justified above).

Now, if you will apply curve fitting techniques to 1st plot you will obtain a fitted polynomial but you will never obtain desired distance from it.

I hope you are getting my point. 

1st plot is combination of various relation between velocity and time of a specific traction system. Initially, a train runs at constant acceleration for sometime, then, at constant power, then after sometime, it attains a constant speed and before stopping,  a constant deceleration after braking is applied. You can observe this clearly in 1st plot.

I will suggest you to go through some texts regarding traction system and hope you find my suggestion valuable. If still doubt remains do contact me.

With Regards 

T. Prakash

The velocity profile is usually given in such a way that there is slow acceleration followed by a constant velocity cruise and finally a slow deceleration which brings the vehicle to a stop. The startup and stop phases are generally modeled through polydyne curves. You need to specify the cruise velocity and the transition time (time taken to accelerate from 0 to cruise velocity and vice versa during braking) to define the polydyne curve equation. You may as well use other polynomial fits like 4-point Bezier or B-spline curves with zero slope at the start and end of the displacement profile.

With closed form expressions for the transition regime (acceleration and braking), you add a constant dwell period in between (the cruise speed). The displacement profile is a consequence of integration of the velocity profile and thus is not an independent graph as shown by you. You need to specify one of the two.

For defining the complete equation for three regimes with polydyne fits you also would need to use switching or heaviside functions which combine the three distinct equations into one. With Bezier fit, you can directly get a single equation. For further details, see the book "Trajectory Planning for Automatic Machines and Robots" by Luigi Biagiotti, Claudio Melchiorri. Specifically, read Chapter 7. You can read parts of this book through google books or download it from Springerlink.

Otherwise, read user manual of VI-Rail or Adams-Car software which deal with vehicle dynamics and traction system modelling.

Hopefully, this information will be useful to you.

thank yuo very much

Tapan Prakash and Arun Kumar Samantaray for your valuable informations

Agree with Tapan Prakash