Time for Non-Uniform Acceleration

[classi]

I've been trying to figure out this seemingly simple problem for a theoretical project I'm doing, but can't figure it out.

Homework Statement

This isn't real, but say that a vehicle travels from 0 to 200 feet linearly at a non-constant acceleration of 32.17 m/s^2 (1g), but the acceleration is in a linear y=mx line. Ignoring friction forces, how long did it take to travel that 200 feet?

Homework Equations

The Attempt at a Solution

I know I can't use kinematics because it's a non-constant acceleration, so I'll probably have to use calculus and differential equations. I know that the graph of velocity will be exponential, and acceleration will be linear. I don't know any times so I can't take the integral.

I know: x1=0, x2=200, a1=0, a2=32.17, but that's about it.

Any ideas on how I can tackle this problem without a function?

 

[Mastermind01]

I think it might help if you stated the problem. The acceleration varies linearly with what? displacement ? time? If a₁ is zero why will the body start moving? Does it have a velocity?