Speed is the absolute value of velocity: speed = \left| v\left( t \right) \right| .

This is the definition of speed, but hardly enough to be sure students know about speed and its relationship to velocity and acceleration.

Velocity is a vector quantity; that is, it has both a direction and a magnitude. The magnitude of velocity vector is the speed. Speed is a non-negative number and has no direction associated with it. Velocity has a magnitude and a direction. Speed has the same value and units as velocity; speed is a number. 

The question that seems to trouble students the most is to determine whether the speed is increasing or decreasing. The short answer is

Speed is increasing when the velocity and acceleration have the same sign.

Speed is decreasing when the velocity and acceleration have different signs.

You should demonstrate this in some real context, such as driving a car (see below). Also, you can explain it graphically.

The figure below shows the graph of the velocity v\left( t \right) (blue graph) of a particle moving on the interval 0\le t\le f. The red graph is \left| v\left( t \right) \right|, the speed. The sections where v\left( t \right)<0 are reflected over the x-axis. (The graphs overlap on [b, d].) It is now quite east to see that the speed is increasing on the intervals [0,a], [b, c] and [d,e].

Another way of approaching the concept is this: the speed is the non-directed length of the vertical segment from the velocity’s graph to the t-axis. Picture the segment shown moving across the graph. When it is getting longer (either above or below the t-axis) the speed increases.

Thinking of the speed as the non-directed distance from the velocity to the axis makes answering the two questions below easy:

    1. What are the values of t at which the speed obtains its (local) maximum values? Answer: x = a, c, and e. 
    2. When do the minimum speeds occur?  What are they? Answer: the speed is zero at b and d

Students often benefit from a verbal explanation of all this. Picture a car moving along a road going forwards (in the positive direction) its velocity is positive.

  • If you step on the gas, acceleration pulls you in the direction you are moving and your speed increases. (v > 0, a > 0, speed increases)
  • Going too fast is not good, so you put on your brakes, you now accelerate in the opposite direction (decelerate?), but you are still moving forward, but slower. (v > 0, a < 0, speed decreases)
  • Finally, you stop. Then you shift into reverse and start moving backwards (negative velocity) and you push on the gas to accelerate in the negative direction, so your speed increases. (v < 0, a < 0, speed increases)
  • Then you put on the breaks (accelerate in the positive direction) and your speed decreases again. (v < 0, a > 0, speed decreases)

Here is an activity that will help your students discover this relationship. Give Part 1 to half the class and Part 2 to the other half. Part 3 (on the back of Part 1 and Part 2) is the same for both groups.  – Added 12-19-17

Also see: A Note on Speed for the purely analytic approach.

Update: “A Note on Speed” added 4-21-2018


7 thoughts on “Speed

  1. Pingback: Adapting 2021 AB 2 | Teaching Calculus

  2. Just one small nitpick, although perhaps it’s my monitor… in the first graph, to me it looks like the graph of the velocity is mostly in orange, not red (only the part where b<x<d, where the velocity and speed overlaps, seems to be red.)


  3. On the AP test, can students just draw a graph of speed (as the absolute value of velocity) and then notice that speed is increasing or decreasing?


    • Twyla
      Good question.
      AP readers are loathe to have to interpret a student’s graph. In these questions they want to see the value of the acceleration and velocity at the time given. If not the value then some indication as to their signs that the student has worked out. If they have the same sign, then the speed is increasing, if they have different signs then decreasing. Sorry.


  4. Hi Lin,
    I do like your approach with the reflection of the graphs!!!

    I also like to discuss with my students, using the graph, the acceleration of the function. You alluded to this in your example.

    Speed would be increasing when the velocity and acceleration have the same sign…

    Looking at the graph we would see where the function was below the axis and where the slope of the tangent line was negative.
    Looking at the graph we would see where the function was above the axis and where the slope of the tangent line was positive.

    (note: stress using endpoints “brackets” when discussing increasing/decreasing)

    Thanks again for sharing!

    Paul Ross


    • Thanks Paul. I just added a Discussion/worksheet about speed to the “Resources” pages (click tab at top of page). I think I meant to include it with the original post.


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