In casual English, "speed," "velocity," and "acceleration" sound interchangeable. In physics, they mean three specific (and different) things. Getting them straight is the first step in any kinematics or motion problem.
Speed: just how fast
Speed is a scalar — a single number with no direction. The speedometer in your car displays speed.
If you're driving 65 mph, that's your speed. The car could be going north, south, in circles, or up a hill. Speed is just magnitude.
Formula: speed = distance ÷ time.
Units: m/s, mph, km/h, knots.
Velocity: how fast AND which direction
Velocity is a vector — magnitude AND direction. It includes the heading.
"60 mph" is speed. "60 mph north" is velocity.
The distinction matters because:
- A car going around a circular track at constant speed has changing velocity (the direction is rotating).
- Two cars at 60 mph in opposite directions have the same speed but opposite velocities.
- Velocity can be negative (moving backward in your reference frame).
Acceleration: how velocity changes
Acceleration is also a vector — the rate of change of velocity.
Formula: acceleration = change in velocity ÷ time.
Units: m/s² (per second squared because velocity itself is per-second, and you're measuring change-per-second-of-that).
Acceleration can be:
- Positive: speeding up in your direction of motion. (Car flooring it.)
- Negative: slowing down. (Car braking.)
- Sideways: changing direction without changing speed. (Car turning a corner at constant 30 mph experiences sideways acceleration.)
Even a car going around a track at constant 60 mph is accelerating — sideways. The velocity vector is rotating, and any change in velocity is acceleration.
The three together: a worked example
You drive from home to work, 10 miles, in 20 minutes. Then home in 25 minutes (rush hour).
Speed: outbound (10 mi / 20 min) = 30 mph average. Return (10 mi / 25 min) = 24 mph average.
Velocity: outbound 30 mph east. Return 24 mph west. Different velocities.
Acceleration: not constant. You accelerate from 0 at home, decelerate at red lights, accelerate again, decelerate to a stop at work. Average acceleration over the trip is hard to define; instantaneous acceleration changes constantly.
Common physics confusion: constant speed isn't constant velocity
A roller coaster cart traveling at constant 50 mph through a loop has constant speed but its velocity is changing all the time (direction rotating). It's accelerating throughout the loop, even though it's not speeding up or slowing down.
This is why physicists distinguish them. The math of motion needs the vector form (velocity, acceleration), not just magnitudes.
Average vs instantaneous
Two more layers:
Average velocity: change in position ÷ time elapsed. Over a long trip, your average velocity is total displacement ÷ total time.
Instantaneous velocity: velocity at a specific moment. The speedometer reading right now.
For straight-line constant motion, they're the same. For varying motion, they differ.
The physics formulas connecting them
For constant acceleration in a straight line:
- v = u + at (final velocity = initial velocity + acceleration × time)
- d = u × t + 0.5 × a × t² (distance = initial velocity × time + half × accel × time²)
- v² = u² + 2ad (final velocity squared = initial squared + 2 × accel × distance)
These are taught in every introductory physics class. They follow from calculus (acceleration is the derivative of velocity, etc.) but work without it for constant-acceleration problems.
Real-world examples
0–60 mph in 6 seconds: change in velocity = 60 mph = 26.8 m/s. Acceleration = 26.8/6 = 4.5 m/s² ≈ 0.46g.
Free fall (skydiver from a plane): g = 9.81 m/s² downward. After 5 seconds: velocity = 0 + 9.81 × 5 = 49 m/s ≈ 110 mph. (Ignoring air resistance.)
Car turning a 100 m radius corner at 30 mph: sideways acceleration = v² / r = (13.4)² / 100 = 1.8 m/s² ≈ 0.18g. Felt as moderate sideways force.
Why this distinction matters
In physics class:
- Speed problems use distance and time only. Easy.
- Velocity problems care about direction. Harder.
- Acceleration problems require derivatives or kinematic equations.
In real life:
- Driving: "speed" is the practical concept. Cops measure speed.
- Aviation: "velocity" matters because heading is critical. Pilots think in vectors.
- Engineering: "acceleration" determines what materials and structures can survive a sudden force.
Calculate them
Our velocity calculator handles speed/velocity from distance and time. The acceleration calculator handles acceleration from velocity changes. Use both for any motion problem.