# 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:

## 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:

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

$\blacksquare$