A little room to wonder

Measuring Motion & Speed

Define the interval, distinguish distance from displacement, and calculate a whole-journey average correctly.

Why can a moving object finish with zero average velocity?

A Newton’s cradle with five metal balls suspended from a frame.
The approved Newton’s-cradle image is a motion context, not a record of the fictional robot’s distances or times.AntHolnes, via Wikimedia Commons. CC BY-SA 3.0. · Source · CC BY-SA 3.0

Decide what was measured

Average speed is total distance travelled divided by elapsed time. Average velocity uses displacement, the change from starting to ending position, divided by elapsed time. Speed has no direction; velocity includes direction. A round trip can have a positive distance while ending at the starting position.

A measurement description needs start and end events, a distance or position reference, units and a timing rule. A single average does not describe every moment between those events. An object might pause or change speed while keeping the same whole-interval average.

Worked example: a fictional robot’s return trip

An imaginary robot travels 3 m along a straight line in 2 s, then returns the same 3 m in 4 s. The first leg’s average speed is 3/2 = 1.5 m/s. The second is 3/4 = 0.75 m/s. For the complete trip, distance is 6 m and elapsed time is 6 s, giving average speed 1 m/s.

Its final position equals its initial position, so displacement and whole-trip average velocity are zero. This does not mean it stayed still. Simply averaging 1.5 and 0.75 gives 1.125 m/s, which is wrong for this trip because the two speeds describe unequal time intervals.

Keep the boundary consistent

The complete trip includes both legs and any time between the chosen start and finish. If the robot had a two-second pause before returning, total elapsed time would become 8 s and average speed would be 6/8 = 0.75 m/s. Omitting the pause would answer a different question.

A position-time graph shows position at recorded times. Its slope over an interval gives average velocity in a one-dimensional model. Distance travelled cannot always be recovered from only the first and last positions: an unrecorded reversal can add distance without changing those endpoints.

Report useful precision

Timing and distance estimates have limits. If a fictional distance were rounded to the nearest metre, reporting a speed to many decimal places would suggest precision the record does not support. Include the measurement description instead of making the number look more certain.

Use the supplied imaginary journey rather than timing people, traffic or animals. A paper ledger can list each segment’s distance and time, then add totals before division. This method keeps the quantity, interval and meaning of the average connected.

Only if you feel like trying

Build the robot’s journey ledger

  1. Write separate distance and elapsed-time entries for both supplied legs.
  2. Calculate whole-trip speed and average velocity, then compare with the incorrect simple mean of leg speeds.
  3. Add the fictional two-second pause and explain exactly which total changes.

Optional. Use paper or talk through the supplied fictional example. No purchase, personal information, account, public sharing or practical experiment is needed.