kinematics, integrate v to x
- Integrate velocity $v$ to obtain position $x$ $$\tag{F1} x(t) - x(t_0) = \int_{t_0}^t \left[ v_0 + \int_{t_0}^{\tau_2} a(\tau_1) d\tau_1 \right] d\tau_2. $$
- Initial condition $x(t_0) = x_0$, will give $$\tag{F2} x(t) = x_0 + v_0 (t - t_0) + \int_{t_0}^t \int_{t_0}^{\tau_2} a(\tau_1) d\tau_1 d\tau_2. $$