Instantaneous velocity

We have defined the average velocity over some time interval as the displacement (change in position) divided by the time interval. 

$$\langle\overrightarrow{v}\rangle =\frac{\Delta\overrightarrow{r}}{\Delta{t}} $$

If the velocity is not uniform, it helps us to talk about the rate of change of position at a particular time -- the instantaneous velocity.

If you are given a function of time, $x(t)$, you learned in calculus how to find its rate of change: just take the derivative.  We can write

$$v(t) =\frac{dx(t)}{dt} $$

But this is a little tricky to think about.  How can we define the velocity at a single instant when our conceptualization of the meaning of velocity required us to compare two positions at two different times?

A reasonable way to do this is to consider a small enough time interval so that the object is (approximately) in uniform motion during that time interval.  We can then define "the velocity at the instant at the center of the time interval".  If we consider a time t and a small time interval Δt, then we'll split the time interval in two and go half way on either side to get the velocity:

$$v(t) =\frac{x(t +\frac{\Delta t}{2}) - x(t -\frac{\Delta t}{2})}{dt} $$

Let's take a specific example: three successive images from a video of the PhET's Moving Man simulation as shown below. The time is shown on the frame and the position (in meters) can be read off the meter stick below him.

If we want his velocity at $t = 3.2 \mathrm{s}$, Our $\Delta t$ here is 2 seconds. 

We have consider two movie frames — the one just before the time we are considering (at $t - \frac{Δt}{2} = 3.1\;\mathrm{s}$) and one just after the time we are considering (at $t + \frac{Δt}{2} = 3.3\;\mathrm{s}$). 

In general, we want our $Δt$ to be very small — smaller than any time we care about — but not TOO small. Especially in biological systems, when we look at our position curves closely, we see that the positions actually fluctuate rapidly due to interaction with jittery molecules in our object's environment or because the mechanism of motion may be in steps. (See for example the Listeria Motion Problem.)

Thinking Graphically

A good way to think both about average and instantaneous velocities is by looking at a graph. The average velocity is given by the change in position divided by the amount of time to make that change — so it's the ratio of the two sides of little triangles attached to the curve — so it's the slope of the hypotenuse of the little triangle (drawn in blue) shown in the figure at the left.  The instantaneous velocity is the ratio of those sides as the triangle gets small — so it's the slope of the tangent to the curve (drawn in red) at the instant we are considering.

Joe Redish and Julia Gouvea 12/30/14