kinematics equations, assumptions

• Integrate acceleration $a$ twice to obtain velocity $v$ and position $x$ $$\tag{D1} v(t) - v(t_0) = \int_{t_0}^t a(\tau) d\tau, $$ $$\tag{D2} x(t) - x(t_0) = \int_{t_0}^t v(\tau) d\tau. $$
• Initial conditions are $v(t_0) = v_0$, $x(t_0) = x_0$, and $t_0 = 0$.