Spring Powered Dragster. Time dependent displacement function

[hammeraxe]

I am to design a gearbox for a small spring powered dragster. It needs to complete a track of 10 m, so I want the spring to give out it's energy over this distance, but without causing wheel slipping.
I also need to produce graphs: acceleration vs time, velocity vs time and distance vs time


The basic layout of the gearbox is clear to me, I just need to choose the right total gear ratio (TGR). To do this, however, I need to be able to evaluate the time in which the vehicle will cover the 10m track and thus choose the quickest combination that does not result in wheel slipping.

A cord is attached to the spring wound around an axis, which, in turn is connected to the wheel axis through gears.

Ignoring wheel moment of inertia, gearbox friction and air resistance:

$$"Wheel torque"=\frac{"Spring force"*R_{axis}}{TGR}$$

$$"Wheel torque"=\frac{"Tractive force"}{R_{wheel}}$$

$$"Tractive force"=\frac{"Spring force"*R_{axis}}{TGR*R_{wheel}}$$

So
$$Acceleration=\frac{"Spring force"*R_{axis}}{TGR*R_{wheel}*"Vehicle mass"}$$

Spring force=k*\Delta*x

I can work out the distance the vehicle moves per unit of spring extenstion decrease (delta x), and it would give me acceleration as a function of distance. This can be integrated and velocity and displacement functions can be obtained.

How do I get time dependence though? I'm sure it's through some sort of differatial manipulation.

I hope I've made this clear. Thanks in advance.

P.S. Sorry about the messy equations, I can't get them to display correctly for some reason

 

[Navin A S]

. To solve for time dependence, you can use the equation:Time=\int_{x_0}^{x_1} \frac{dx}{v(x)}where x_0 and x_1 are the initial and final positions of the vehicle, and v(x) is the velocity of the vehicle at position x. Integrating this equation gives you the total time it takes for the vehicle to travel between its initial and final positions. From here, you can calculate the acceleration, velocity, and distance as a function of time.