Motion Graphs and Calculus

We know that velocity is change in displacement, and acceleration is change in velocity. Calculus tells us that the rate of change of a function is the derivative, and therefore we can differentiate and integrate to convert between different functions of displacement, velocity and acceleration.

graph LR
	A(Displacement) --->|Differentiation| B(Velocity) --->|Differentiation| C(Acceleration)
	C --->|Integration| B --->|Integration| A

Formulae

v = \frac{d s}{d t} a = \frac{d v}{d t}

v = \int a d t s = \int v d t

Variable Key

$s$ is displacement.
$v$ is velocity.
$a$ is acceleration.
$t$ is time.