Speed
Speed is the rate of motion, or equivalently the rate of change in position, often 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:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): v = \left|\frac {d}{t}\right|.
Note that "v" is the variable for speed.
Objects that move horizontally as well as vertically (such as aircraft) distinguish forward speed and climbing speed.
Units of speed include:
- Mach 1 ≈ 343 ms-1 ≈ 1235 km/h ≈ 768 mph in dry air at sea-level pressure and 293 kelvin (See Speed of sound for more detail.)
- c = 299,792,458 ms-1
- Other important conversions
- 1 m/s = 3.6 km/h
- 1 mph = 1.609 km/h
- 1 knot = 1.852 km/h = 0.514 ms-1
Vehicles often have a speedometer to measure the speed they are going.
[edit] Average speed
Speed as a physical property represents primarily instantaneous speed. In real life we often use average speed (denoted Failed to parse (Missing texvc executable;
please see math/README to configure.): |\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:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\tilde{v}| = \frac{\Delta l}{\Delta t}
Instantaneous speed defined as a function of time on interval Failed to parse (Missing texvc executable;
please see math/README to configure.): [t_0, t_1]
gives average speed:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\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 Failed to parse (Missing texvc executable;
please see math/README to configure.): [l_0, l_1]
gives average speed:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\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 Failed to parse (Missing texvc executable;
please see math/README to configure.): |v|_{a}
and second half with speed Failed to parse (Missing texvc executable;
please see math/README to configure.): |v|_{b}
, produces total average speed Failed to parse (Missing texvc executable;
please see math/README to configure.): |\tilde{v}| = \frac{|v|_a + |v|_b}{2}
. The correct value is Failed to parse (Missing texvc executable;
please see math/README to configure.): |\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):
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |v| \sim D_t\; \Rightarrow \; |\tilde{v}| = \int |v| D_t(|v|) \, dv
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |v| \sim D_l\; \Rightarrow \; |\tilde{v}| = \frac{1}{\int \frac{D_l(|v|)}{|v|} \, dv}
[edit] 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 ms-1; 0.0036 km/h; 0.0023 mph.
- A brisk walk = 1.667 ms-1; 6 km/h; 3.75 mph.
- Olympic sprinters (average speed over 100 metres) = 10 ms-1; 36 km/h; 22.5 mph.
- Speed limit on a French autoroute = 36.111 ms-1; 130 km/h; 80 mph.
- Top cruising speed of a Boeing 747-8 = 290.947 ms-1; 1047.41 km/h; 650.83 mph; (officially Mach 0.85)
- Official air speed record = 980.278 ms-1; 3,529 km/h; 2,188 mph.
- Space shuttle on re-entry = 7,777.778 ms-1; 28,000 km/h; 17,500 mph.
- the speed of sound in air (Mach 1) is about 340 ms-1, and 1500 ms-1 in water
- Taipei 101 Observatory Elevator = 1010 m/min ; 16.667 ms-1 ; 60.6 km/h; 37.6 mph
[edit] See also
[edit] External Links
[edit] References
- Weisstein, Eric W. "Speed." From MathWorld--A Wolfram Web Resource. [1]
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