Body under Constant Acceleration/Velocity after Time

Theorem
Let $B$ be a body under constant acceleration $\mathbf a$.

Then:
 * $\mathbf v = \mathbf u + \mathbf a t$

where:
 * $\mathbf u$ is the velocity at time $t = 0$
 * $\mathbf v$ is the velocity at time $t$.

Proof
By definition of acceleration:
 * $\dfrac {\mathrm d \mathbf v} {\mathrm d t} = \mathbf a$

By Solution to Linear First Order Ordinary Differential Equation:
 * $\mathbf v = \mathbf c + \mathbf a t$

where $\mathbf c$ is a constant vector.

We are given the initial condition:
 * $\big.{\mathbf v}\big\rvert_{\, t \mathop = 0} = \mathbf u$

from which it follows immediately that:
 * $\mathbf v = \mathbf u + \mathbf a t$