calculus-based physics for scientists and engineers

The Velocity Function

The Velocity Function in One Dimension

The velocity function is the time-derivative of the position function.

Repeat this seven times. Memorize this. There is an infinite number of mathematical representations of this relationship. However, the fundamental relationship between the velocity and the position is the same: The function that describes the velocity of an object is the derivative (with respect to time) of the function that describes the position of the object.

The Graphical Representation of Velocity and Position

That this gets its very own topic section should be a clue. Since the velocity is the derivative of the position, it is the slope of the tangent line of the position function (remember that from calculus!). You can use this information to extract information from their graphs.

Average Speed and Average Velocity

Let's look at the differences between average speed and average velocity. Spoiler alert! They are not the same thing. This topic contains a meaningful fraction of all trick questions in physics.


Do Now!   Do these exercises immediately.

Not Now!   Do these after you start to forget the topic, say in a week.

More!   More exercises if you want. Maybe review before a test.

Required Problems


1. You drive 30 miles at 30 mph. How fast do you need to drive the next 30 to average 60 mph?

Optional Problems