Non-Constant Acceleration

For motion in a straight line, define:

• $x(t)$: displacement
• $v(t)$: velocity
• $a(t)$: acceleration

Fundamental relationships

Velocity is the rate of change of displacement:

$$v = \frac{dx}{dt}$$

Acceleration is the rate of change of velocity (which is the second derivative of displacement with respect to time):

$$a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

Reversing these formulae

Since to get acceleration we differentiate velocity, to get velocity we can integrate acceleration.

$$v(t) = \int a(t)\,dt + C$$

The same principle applies to displacement and velocity

$$x(t) = \int v(t)\,dt + C$$