CourseWiki/Engineering Fundamentals/Motion and Rotation

Motion Quantities

Distinguish path length from displacement, and speed from velocity, before using time to describe a machine or vehicle in motion.

Path length and displacement answer different questions

Distance is how much path was traveled. Displacement is the directed change from the starting position to the ending position. A return trip can add distance while leaving displacement at zero.

Distance: A nonnegative scalar path length. A conveyor that moves a tote 12 m out and 12 m back travels 24 m of distance.
Displacement: A change in position with direction. If the tote returns to its start, its displacement is 0 m even though it moved.
Coordinate choice: Choose a positive direction before calculating. Along a straight robot rail, right might be positive and left negative.

Distance tells what the machine did along its path; displacement tells where it ended up relative to where it began. Do not substitute one for the other just because both use meters or feet.

Speed and velocity are not synonyms

Average speed is total distance divided by elapsed time. Average velocity is displacement divided by elapsed time. Velocity includes direction; speed does not.

average speed = total distance / elapsed time
average velocity = displacement / elapsed time

Notation bridge: this site uses x for position and Δx for displacement. Some course materials use s for position instead. For a stated interval, write average velocity as v_avg = Δx / Δt; the symbols differ, but the physical idea is the same.

For the 12 m out-and-back path completed in 16 s, average speed is 24 m / 16 s = 1.5 m/s. Average velocity is 0 m / 16 s = 0 m/s. Both statements can be true at the same time.

A production-line example

A mechanism must carry a part 12 m in 8 s. Its required average speed is 12 m / 8 s = 1.5 m/s. Converting with exact factors gives 1.5 m/s × (39.37007874 in / 1 m) = 59.055... in/s, or 59.1 in/s. If the programmed direction is stated, its average velocity is +59.1 in/s in that direction.

Average and instantaneous readings

An average uses a time interval. An instantaneous reading describes one moment: a car speedometer reports instantaneous speed; a robot controller’s live velocity value is an instantaneous velocity in its coordinate frame.

Use average speed for throughput, travel-time, and trip planning.
Use average velocity for a net movement over a stated interval.
Use an instantaneous reading when clearance, collision avoidance, or a machine limit depends on what is happening now.

The calculus idea, after the practical one

Over a smaller and smaller time interval, average velocity approaches instantaneous velocity. In calculus notation, velocity is the derivative of position with respect to time: v = dx/dt. For this course, begin by naming the interval, position change, and direction correctly; the derivative notation is a compact extension of the same idea.

Out-and-back path lab

Set positions along a straight path and the outbound and return times. The text results make the distance and displacement distinction visible even when the track is not used.

Position and time inputs

Start position (m)

Turn position (m)

Outbound time (s)

Return time (s)

Start: 0 m Turn: 12 m End: 0 m

Distance: 24 m
Displacement: 0 m
Average speed: 1.5 m/s
Average velocity: 0 m/s

JavaScript is off. The displayed values are the solved 12 m out, 12 m back, 8 s each-way case: 24 m distance, 0 m displacement, 1.5 m/s average speed, and 0 m/s average velocity.

Practice and sources; Practice

Practice and sources

Use the practice bank to apply distance, speed, and round-trip rate reasoning.

Add to practice

Source note

The NIST SI Units reference lists velocity as a derived unit example. Direction is an added physical requirement: two motions may have the same speed but different velocities because their directions differ.

Appears in

DET 403 DET Units, Motion, and Force