Calculus How To

Speed: Definition, Simple Example

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Common Usage
Calculus Definition


Speed (Common Usage)

In everyday language, “speed” tells you how quickly an object travels between two places.

The formula is as follows:

S = D/T

  • S = Speed
  • D = Distance
  • T = Time

S, D, and T can have different units. For example:

  • S = miles-per-hour (mph) or kilometers-per-hour.
  • D = miles, millimeters or feet. 
  • T = larger intervals like light years or smaller ones like nanoseconds.

Sometimes “velocity” and “speed” are used to mean the same thing. However, there are very important distinctions between the two when used in calculus (or physics).

Calculus Definition

Speed takes on a more precise definition when used in calculus.  It is defined as the absolute value of velocity:

S = │v(t)│


  • t = time

Speed is a scalar quantity represented by magnitude (i.e. an amount) and doesn’t have a direction—unlike velocity, which is a vector with both magnitude and direction.  Speed measures the rate of motion regardless of direction, and is measured in distance per units of time.  As you’re taking the absolute values, it is always positive.

Example: Calculating Speed using Derivatives

We don’t really calculate “speed” in calculus, so if you’re asked a question like “how fast is the object traveling”, what the question is really asking you for is the velocity of the object.

When you’re given a position function, you find velocity with the position function’s first derivative:
s'(t) = v(t)

Example question: Based on the function of s(t) = t3 + 4t feet at time (t), find S at t = 4.24 seconds.

Step 1: Take the derivative of the function s(t) = t3 + 4t:

s'(t) = v(t)

v(t) = 3t2 + 4

Step 2: Plug the given time (in this example, that’s t = 4.24 seconds) into the function you found in Step 1:

v(4.24) = 3(4.24)2 + 4

v(4.24) = 3(17.9776) + 4

v(4.24) = 53.93 + 4

v(4.24) = 57.93

Step 3: Find the absolute value:

S = │v(4.24)│ = 4.24 seconds = 57.93 feet/second

The object is traveling at 57.93 feet per second at 4.24 seconds.


Bauer, W. & Westfall, G. (2011). University Physics. McGraw Hill.
Davis, D. (2002). Retrieved May 16, 2019 from:

Stephanie Glen. "Speed: Definition, Simple Example" From Calculus for the rest of us!

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