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:

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

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:

$\bigvalueat {\mathbf v} {t \mathop = 0} = \mathbf u$

from which it follows immediately that:

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

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equation of motion