Accelerating twice

$a_c = kx_p$ , where $a_c$ is acceleration of the car, k is a constant, and $x_p$ is position of the pedal.

$\therefore \frac {d^2a}{dt^2}=ka_p=constant$

$\frac {da}{dt}$ $\alpha$ $t$

$a$ $\alpha$ $t^2$

$v$ $\alpha$ $t^3$

$\therefore x_c$ $\alpha$ $t^4$

(Constants of integration are neglected here, as given in the question.)

As stated in the question, all constants can be ignored, so let's assume $a_{pedal} = 1$ and ignore constants of integration. We will use Big O Notation

$a_{pedal} = 1$

$v_{pedal} = \int a_{pedal} \, dt = \int 1 \, dt = t$

$x_{pedal} = \int v_{pedal} \, dt = \int t \, dt = \frac 12 t^2 = O(t^2)$

$x_{pedal} = a_{car} = O(t^2)$

$v_{car} = \int a_{car} \, dt = \int t^2 \, dt = \frac 13 t^3 = O(t^3)$

$x_{car} = \int v_{car} \, dt = \int t^3 \, dt = \frac 14 t^4 = \boxed{O(t^4)}$