What's the relation between speed of an aircraft and takeoff distance?
I think your question is regarding new aircraft under design and not for existing planes?

The dynamics of ground roll/stopping is governed by one equation:

$$m\ddot{x}=T-D-\mu(L-mg)$$

where $T$ is thrust, $D$ is drag, $\mu$ is friction coefficient, $L$ is lift, $m$ is mass and $g$ is gravitational accel. Since thrust, drag and lift are a function of airspeed (and spoilers + brakes + thrust reverser should abort occur), the most naive way is just to numerically integrate the equation twice to get the distance at time.

That being said, don't forget that takeoff distance and accelerate-stop distance must be considered with all engines operating and with the critical engine failed. $V_1$, assuming not limited by controllability, should be selected such that the two distances balance. Also keep in mind that a good portion of takeoff distance is rotating and accelerating to $V_2$ before 35ft with the critical engine failed. The previous equation does not account for that portion.