Body under Constant Acceleration/Velocity after Time

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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 definiton 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:

$\mathbf v = \mathbf u + \mathbf a t$

$\blacksquare$