# Speed

Speed is the rate of motion, or equivalently the rate of change in position, many times expressed as distance d traveled per unit of time t.

Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity.

In mathematical notation, it is simply:

${\displaystyle |v|={\frac {d}{t}}}$

Note that "v" equals velocity.

Objects that move horizontally as well as vertically (such as aircraft) distinguish forward speed and climbing speed.

## Units

Units of speed include:

Mach 1 ≈ 343 m/s ≈ 1235 km/h ≈ 768 mph (see the speed of sound for more detail)
c = 299,792,458 m/s
• Other important conversions
1 m/s = 3.6 km/h
1 mph = 1.609 km/h
1 knot = 1.852 km/h = 0.514 m/s

Vehicles often have a speedometer to measure the speed.

## Average speed

Speed as a physical property represents primarily instantaneous speed. In real life we often use average speed (denoted ${\displaystyle |{\tilde {v}}|}$), which is rate of total distance (or length) and time interval. For example, if you go 60 miles in 2 hours, your average speed during that time is 60/2 = 30 miles per hour, but your instantaneous speed may have varied.

In mathematical notation:

${\displaystyle |{\tilde {v}}|={\frac {\Delta l}{\Delta t}}}$

Instantaneous speed defined as a function of time on interval ${\displaystyle [t_{0},t_{1}]}$ gives average speed:

${\displaystyle |{\tilde {v}}|={\frac {\int _{t_{0}}^{t_{1}}|v|(t)\,dt}{\Delta t}}}$

while instant speed defined as a function of distance (or length) on interval ${\displaystyle [l_{0},l_{1}]}$ gives average speed:

${\displaystyle |{\tilde {v}}|={\frac {\Delta l}{\int _{l_{0}}^{l_{1}}{\frac {1}{|v|(l)}}\,dl}}}$

It is often intuitively expected, but incorrect, that going half a distance with speed ${\displaystyle |v|_{a}}$ and second half with speed ${\displaystyle |v|_{b}}$, produces total average speed ${\displaystyle |{\tilde {v}}|={\frac {|v|_{a}+|v|_{b}}{2}}}$. The correct value is ${\displaystyle |{\tilde {v}}|={\frac {2}{{\frac {1}{|v|_{a}}}+{\frac {1}{|v|_{b}}}}}}$
(Note that the first is a proper arithmetic mean while the second is a proper harmonic mean).

Average speed can be derived also from speed distribution function (either in time or on distance):

${\displaystyle |v|\sim D_{t}\;\Rightarrow \;|{\tilde {v}}|=\int |v|D_{t}(|v|)\,dv}$
${\displaystyle |v|\sim D_{l}\;\Rightarrow \;|{\tilde {v}}|={\frac {1}{\int {\frac {D_{l}(|v|)}{|v|}}\,dv}}}$

## Examples of different speeds

Below are some examples of different speed (see also main article Orders of magnitude (speed)):

• Speed of a common snail = 0.001 m/s; 0.0036 km/h; 0.0023 mph.
• A brisk walk = 1.667 m/s; 6 km/h; 3.75 mph.
• Olympic sprinters (average speed over 100 metres) = 10 m/s; 36 km/h; 22.5 mph.
• Speed limit on a French autoroute = 36.111 m/s; 130 km/h; 80 mph.
• Top cruising speed of a Boeing 747-8 = 290.947 m/s; 1047.41 km/h; 650.83 mph; (officially Mach 0.85)
• Official air speed record = 980.278 m/s; 3,529 km/h; 2,188 mph.
• Space shuttle on re-entry = 7,777.778 m/s; 28,000 km/h; 17,500 mph.
• the speed of sound in air (Mach 1) is about 340 m/s, and 1500 m/s in water
• Taipei 101 Observatory Elevator = 1010 m/min ; 16.667 m/s ; 60.6 km/h; 37.6 mph