Section 2.24 - Derivation: a motion equation via calculus

Motion equations can be derived using calculus as well as algebra. As an example, we derive the equation shown in Equation 1 to the right.

Variables

The final velocity and final position are given by functions of time.

	acceleration	a
	initial velocity	v_i
	(final) velocity	v(t)
	elapsed time	t
	initial position	x_i
	(final) position	x(t)
	displacement	Δx = x(t) − x_i

Strategy

We start with the definition of velocity as the derivative of position with respect to time, and use integration to find an equation for the position function.
Then we find an equation for Δx by subtracting the initial position (when t = 0) from this position function.

Physics principles and equations

We use the definition of velocity as a derivative

v(t) = dx/dt

We also use one of the basic motion equations

v(t) = v_i + at

This equation was stated before with v_f on the left side instead of v(t). But we can compute the final velocity as a function of time by writing the equation in this fashion.

Step-by-step derivation