Velocity
Velocity is the rate of change of displacement with respect to time, making it a vector quantity with both magnitude (speed) and direction. Average velocity equals total displacement divided by total time, while instantaneous velocity is the derivative of position with respect to time. Velocity is central to Newton's laws and is measured in metres per second (m/s).
Key Formula
v_avg = Δx / Δt
LaTeX: \vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t}
| Symbol | Meaning | Unit |
|---|---|---|
| v⃗_avg | Average velocity | m/s |
| Δx⃗ | Displacement | m |
| Δt | Time interval | s |
Worked Example
Problem
A train travels 150 km due north in 2 hours, then 90 km due east in 1 hour. What is the magnitude and direction of the train's average velocity for the entire journey?
Solution
Step 1: Total time = 2 + 1 = 3 h = 10800 s. Step 2: Net displacement: north component = 150 km, east component = 90 km. Step 3: Magnitude of displacement = √(150² + 90²) = √(22500 + 8100) = √30600 ≈ 174.9 km. Step 4: Average velocity = 174.9 km / 3 h ≈ 58.3 km/h. Step 5: Direction = arctan(90 / 150) = arctan(0.6) ≈ 31° east of north.
Answer
Average velocity ≈ 58.3 km/h at 31° east of north.
Speed vs. Velocity — key differences
| Property | Speed | Velocity |
|---|---|---|
| Quantity type | Scalar | Vector |
| Definition | Distance / time | Displacement / time |
| Direction included? | No | Yes |
| Can be zero while moving? | No | Yes (circular path averages to zero) |
| SI Unit | m/s | m/s |
| Symbol | v or s | v⃗ |
Interactive Tools
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Related Terms
Speed
Speed is the rate at which an object covers distance, defined as the total distance travelled divided by the time taken. It is a scalar quantity, possessing magnitude but no direction, and is always non-negative. Average speed is useful for describing overall motion, while instantaneous speed gives the speed at any particular moment.
Acceleration
Acceleration is the rate of change of velocity with respect to time, and is a vector quantity. An object accelerates whenever its speed changes, its direction changes, or both simultaneously. Acceleration is caused by a net force (Newton's second law) and is measured in metres per second squared (m/s²).
Displacement
Displacement is the shortest straight-line distance between an object's initial and final positions, measured as a vector quantity with both magnitude and direction. Unlike distance, displacement does not account for the actual path taken, only the net change in position. It is the fundamental quantity used to define velocity and is measured in metres (m).
From Latin "velocitas" (swiftness, speed), derived from "velox" (swift). The term was adopted into classical mechanics in the 17th century by Isaac Newton and Gottfried Leibniz to describe directional rate of motion.