# What does the slope of a distance-versus-time graph represent?

Jan 1, 2017

The slope of a position-versus-time graph represents the velocity.

#### Explanation:

The slope of the tangent line represents the rate of change of a function. Velocity is the rate of change of position with respect to time. Equivalently, velocity is the derivative of the position function.

${\vec{v}}_{a v g} = \frac{\Delta s}{\Delta t}$

The average velocity has limited usefulness for an object whose velocity is not constant.

We can define an object's instantaneous velocity at a single instant of time, $t$, where the instantaneous velocity at time $t$ is the slope of the line that is tangent to the position-versus-time graph at time $t$.

$\vec{v} = \lim \left(\Delta t \to 0\right) \frac{\Delta s}{\Delta t} = \frac{\mathrm{ds}}{\mathrm{dt}}$