Speed, Velocity, Distance, and Displacement - Vocabulary

Quantities can be scalars or vectors
- Scalar: magnitude only (size or extent)
- Vector: magnitude and direction
Speed is a scalar (magnitude only)
- Examples: a person running at 4 m/s, a plane at 250 m/s
Velocity is a vector (magnitude and direction)
- Examples: a person cycling 6 m/s east, a train moving backwards at 55 m/s
It’s common to confuse speed and velocity in exams; keep track of whether direction is included.

Distance is a scalar (magnitude only)
- Examples: 10 meters, 40 miles
Displacement is a vector (magnitude and direction)
- Examples: 40 meters east, 2 meters downward

Bridge length: 550 meters
Time to cross: 10 seconds
Speed calculation (scalar):
s = \frac{d}{t} = \frac{550\ \text{m}}{10\ \text{s}} = 55\ \text{m/s}
If we use displacement for the same journey (vector):
- Displacement: 550 meters east
- Time: 10 seconds
- Velocity calculation (vector):
  \mathbf{v} = \frac{\mathbf{s}}{t} = \frac{550\ \text{m east}}{10\ \text{s}} = 55\ \text{m/s east}
Key takeaway: magnitude can be the same, but velocity includes direction.

Speed equation: s = \frac{d}{t}
- Here, s is speed, d is distance (scalar), and t is time
Velocity equation: \mathbf{v} = \frac{\mathbf{s}}{t}
- Here, \mathbf{s} is displacement vector, and t is time
The two equations differ in the numerator variable (speed uses distance; velocity uses displacement) but share the same time denominator
In practice, many people use the velocity equation regardless of whether a direction is explicitly given; this yields a velocity with a direction (even if implicit)
Velocity can be negative to represent movement in the opposite direction (e.g., backwards)
If motion isn’t at a constant speed, you can still use the same formulas by using total distance or total displacement divided by total time to get average speed or average velocity

Average speed
- If the journey’s speed varies, average speed is:
  \overline{s} = \frac{D}{T}
- where D is total distance traveled and T is total time
Average velocity
- If the journey’s displacement varies, average velocity is:
  \overline{\mathbf{v}}=\frac{\Delta x}{T}
- where (\Delta \mathbf{r}) is total displacement vector
Note: The average velocity can differ from the instantaneous velocity at any moment

In air, sound travels at about 330 m/s
This speed changes when sound travels through different media (e.g., water vs air)
Remember: sound speed is not constant across all environments

Wind is the natural movement of air and varies widely
- From almost zero m/s on a still day to faster than a speeding train on windy days
Wind speed is affected by factors such as temperature, atmospheric pressure, and structures like buildings or mountains

Always identify whether the quantity is scalar (magnitude only) or vector (magnitude + direction)
Distinguish distance (scalar) from displacement (vector)
Use the appropriate formula and note the numerator variable:
- Speed: s = \frac{d}{t} with $d$ as distance
- Velocity: \mathbf{v} = \frac{\mathbf{s}}{t} with $\mathbf{s}$ as displacement
For non-constant motion, compute average quantities using total distance or total displacement divided by total time
When giving velocity, you can include direction (e.g., “55 m/s east”). If direction is omitted, it is still valid as a magnitude, but you lose directional information
Real-world speeds vary; use standard reference values as rough guides, and remember that factors like health, effort, and environment matter
The speed of sound and its dependence on the medium is an important concept for later topics (e.g., waves and acoustics)

Understanding the distinction between speed, velocity, distance, and displacement and how to apply the correct formula will help avoid common exam mistakes and give you a solid foundation for motion analysis.