How Do You Find Linear Velocity? Formula, Units & Examples
Linear velocity measures how fast an object moves along a path — it’s the rate at which position changes over time. To find linear velocity, divide the distance traveled by the time it took to travel that distance, giving you v = d / t for straight-line motion. If you’re working with a rotating object instead, you use v = ωr, where ω is the angular velocity and r is the radius. Below are the formulas, the units, and a worked example so you can apply it confidently.
The Core Linear Velocity Formula
For motion in a straight line, linear velocity is simply distance divided by time:
v = d / t
Where:
- v = linear velocity
- d = distance traveled
- t = time elapsed
If you include direction, velocity becomes a vector and you’d use displacement instead of distance. Khan Academy treats this distinction carefully: speed is the magnitude, velocity carries direction too. For most everyday problems, the scalar version is what you need.
Average vs. Instantaneous Velocity
Average velocity looks at the total picture — total distance over total time. Instantaneous velocity is the velocity at one exact moment, which you’d find by taking the limit as the time interval shrinks toward zero. On a position-time graph, average velocity is the slope of the straight line connecting two points, while instantaneous velocity is the slope of the tangent line at a single point. The Physics Classroom explains this graphically: a steeper slope means a faster object, and a flat line means the object is stationary.
For constant-velocity problems, average and instantaneous velocity are the same, which is why the simple formula works so well.
Units of Linear Velocity
Linear velocity is expressed as a distance unit divided by a time unit. The SI unit is meters per second (m/s).
| Unit | Common Use |
|---|---|
| m/s | SI standard, science and engineering |
| km/h | Road speeds, everyday contexts |
| ft/s | US engineering, some sports analysis |
| mph | US and UK road speeds |
To convert between them: 1 m/s equals 3.6 km/h, and 1 m/s equals about 2.237 mph. So a car moving at 25 m/s is doing 90 km/h.
The key is consistency — if distance is in meters, time must be in seconds, or your units won’t cancel correctly and your answer will be wrong.
Linear Velocity in Circular Motion
When an object moves in a circle, its linear velocity is tangent to the circle at every point. That’s the direction it would fly off if the centripetal force vanished. The formula here connects linear and angular motion:
v = ωr
Where:
- v = linear velocity
- ω (omega) = angular velocity, in radians per second
- r = radius of the circle
OpenStax makes a useful point: every point on a rigid rotating object shares the same angular velocity, but points farther from the center have greater linear velocity. That’s why the outer edge of a spinning wheel moves faster than a point near the hub.
Worked Example: Circular Motion
A bicycle wheel has a radius of 0.35 m and spins at 4 radians per second.
v = ωr = (4 rad/s)(0.35 m) = 1.4 m/s
That’s the linear speed of a point on the tire’s edge. If you doubled the radius, the speed would double too — same rotation, faster travel.
Turning Angular Speed into Linear Speed
Sometimes you’re given revolutions per minute (rpm) rather than radians per second. Convert first, then plug in.
- Convert rpm to revolutions per second by dividing by 60.
- Convert revolutions to radians by multiplying by 2π (since one full revolution is 2π radians).
- Multiply by radius to get linear velocity.
For example, a wheel spinning at 300 rpm:
- 300 ÷ 60 = 5 revolutions per second
- 5 × 2π ≈ 31.4 rad/s
- With a radius of 0.2 m: v = 31.4 × 0.2 ≈ 6.28 m/s
This conversion chain shows up constantly in problems involving gears, pulleys, and rotating machinery, so it’s worth committing to memory.
Common Mistakes to Avoid
- Mixing units. Converting time to minutes but leaving distance in meters breaks the calculation. Convert everything to matching units first.
- Confusing linear and angular velocity. They’re related but distinct — angular velocity describes rotation rate, linear velocity describes actual travel speed along the path.
- Using diameter instead of radius. The formula v = ωr needs the radius, not the full width across the circle.
- Ignoring direction when it matters. For vector problems, velocity must include a direction; otherwise you’re only reporting speed.
Checking your units at the end is a quick way to catch most of these errors before they cost you points.
Key Takeaways
- For straight-line motion, find linear velocity with v = d / t (distance divided by time).
- For circular motion, use v = ωr, where ω is angular velocity in rad/s and r is the radius.
- The SI unit is meters per second (m/s); convert rpm to rad/s before using v = ωr.
- Average velocity uses total distance over total time; instantaneous velocity is the slope of a position-time graph at one point.
- Points farther from the center of a rotating object have greater linear velocity but the same angular velocity.
- Always match your units before calculating, and include direction when working with vectors.
