Speed and Velocity

Speed and Velocity: The Physics of Motion

In common speech, "speed" and "velocity" are used interchangeably. In Physics, they are fundamentally different because one cares about direction and the other does not.

1. Speed ($v$)

Speed is a scalar quantity. It is the rate at which an object covers distance.

Proportionality:

Speed is directly proportional to distance ($v \propto s$) when time is constant.
Speed is inversely proportional to time ($v \propto 1/t$) when distance is constant.

The Basic Formula:

$$v = \frac{s}{t}$$

Average vs. Instantaneous Speed

Average Speed ($\bar{v}$)

The total distance divided by the total time. It ignores the stops and starts in between.

$$\bar{v} = \frac{\Delta s}{\Delta t}$$

Instantaneous Speed

The speed at a specific moment (what your car's speedometer shows). Mathematically, it is the limit as time approaches zero:

$$v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$$

2. Velocity ($\mathbf{v}$)

Velocity is a vector quantity. It is the rate of change of displacement. While speed asks "How fast?", velocity asks "How fast and in what direction?"

Average Velocity ($\mathbf{\bar{v}}$):

$$\mathbf{\bar{v}} = \frac{\Delta \mathbf{s}}{\Delta t}$$

The Magnitude Connection

The magnitude of instantaneous velocity is exactly equal to the instantaneous speed ($v = |\mathbf{v}|$).

3. Units and Conversions

The SI unit for both is meters per second (m/s).

Common Conversion Rules of Thumb:

Unit	Metric Conversion	Quick Estimation
1 km/h	$\approx 0.28 \text{ m/s}$	$\approx 1/4 \text{ m/s}$
1 mph	$\approx 0.45 \text{ m/s}$	$\approx 1/2 \text{ m/s}$
1 knot	$1 \text{ nautical mile/h}$	$\approx 0.51 \text{ m/s}$

4. Special Speed Constants

In advanced physics and IJSO theory, we often measure speed relative to universal constants:

Mach Number: The ratio of an object's speed to the speed of sound ($\approx 343 \text{ m/s}$ at room temp).
Example: The Concorde flew at Mach 2.0 ($2 \times$ speed of sound).
Speed of Light ($c$): The ultimate speed limit of the universe.
$c = 299,792,458 \text{ m/s}$ (Commonly used as $3.00 \times 10^8 \text{ m/s}$).

5. Speed Benchmarks (Slowest to Fastest)

To give your students perspective, here is a "Speed Scale" for your site:

Speed (m/s)	Phenomenon
$10^{-9}$	Continental plate movement / Hair growth
$1.3$	Typical human walking pace
$10.4$	Usain Bolt (World Record Sprint)
$343$	Speed of sound (Air, Room Temp)
$7,700$	International Space Station (ISS) in orbit
$299,792,458$	Speed of light in a vacuum ($c$)