CourseWiki/Engineering Fundamentals/Motion and Rotation

Acceleration and Squared Time

Calculate change in velocity over time, distinguish average from instantaneous acceleration, and convert squared time units correctly.

Acceleration is change in velocity over time

Acceleration is not simply “speeding up.” It is a change in velocity, so it can describe speeding up, slowing down, or turning.

a_avg = Δv / Δt = (v_f − v_i) / (t_f − t_i)

Δv: Change in velocity. With one chosen positive direction, Δv = v_f − v_i.
m/s²: Meters per second per second. Each second, velocity changes by the stated number of meters per second.
ft/s²: Feet per second per second. The coherent foot–slug–second system uses this acceleration unit with slugs for mass.

Write the sign. A final velocity lower than the initial velocity produces negative acceleration in the selected direction; it is not an error just because the answer is negative.

Average is an interval; instantaneous is a moment

Average acceleration is calculated over stated start and end conditions. An accelerometer or control system can report instantaneous acceleration at one moment. They match at every moment only when the acceleration stays constant through the interval.

Machine arm example

A machine arm starts from rest and reaches 3 m/s in 4 s:

a_avg = (3 m/s − 0 m/s) / 4 s = 0.75 m/s².

This is the average acceleration for the four-second move. If the drive ramps gently at the start and end, its instantaneous acceleration changes during the move even though the average remains 0.75 m/s².

Robot gripper example

A SolidWorks motion study programs a gripper from 0 to 5 ft/s in 0.5 s to pick an electronic component. Its average acceleration is 5 ft/s / 0.5 s = 10 ft/s². In SI, 10 ft/s² × (0.3048 m / 1 ft) = 3.048 m/s².

Squared time needs a squared factor; Watch out

Squared time needs a squared factor

Acceleration has time in the denominator twice. When converting an acceleration unit, convert the entire squared time relationship, not just one s.

1 min = 60 s means 1 min² = (60 s)² = 3,600 s².

Use the inverse relationship in the correct direction. 1 ft/s² = 0.3048 m/s², not 0.3048² m/s². The foot is the only unit being changed in that expression; seconds are already seconds in both units.

By contrast, converting m/min² to m/s² requires the full squared time factor. The unit cancellation shows why:

1 m/min² × (1 min / 60 s)² = 1/3,600 m/s².

Acceleration lab

Change the initial and final velocities and the elapsed time. The last result converts the calculated foot-per-second-squared value to SI.

Velocity change

Initial velocity (ft/s)

Final velocity (ft/s)

Elapsed time (s)

Change in velocity: 5 ft/s
Average acceleration: 10 ft/s²
SI equivalent: 3.048 m/s²

Practice and sources; Practice

Practice and sources

Use the practice bank to apply average acceleration over a stated interval.

Add to practice

Source note

NASA’s Newton’s second-law resource connects force, mass, and acceleration for grades 9–12. The calculation here uses the average velocity-change form first; a real machine profile may have changing instantaneous acceleration within the same interval.

Appears in

DET 403 DET Units, Motion, and Force