Question

Write dawn the relationship between acceleration, initial velocity and final velocity.

Answer 1

By definition, the acceleration is (bold letters denote vectors):

$$\mathbf{a} = \dfrac{d\mathbf{v}}{dt}$$

where $\mathbf{v}$ is the instantaneous velocity, and $t$ is the time. Thus, to find the final velocity at some moment of time one should integrate acceleration with respect to $t$:

$$\mathbf{v}_f = \mathbf{v}_0 + \int_{t_0}^t\mathbf{a}dt$$

where $\mathbf{v}_f$ is the final velocity, $\mathbf{v}_0$ is the initial velocity, $t_0, t$ are initial and final moments of time respectively.

Answer. $\mathbf{v}_f = \mathbf{v}_0 + \int_{t_0}^t\mathbf{a}dt$.