Speed

From Exampleproblems

For alternate uses, see special education or speed (disambiguation).

Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved 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.

Units of speed include:

metres per second, (symbol m/s), the SI derived unit
kilometres per hour, (symbol km/h)
miles per hour, (symbol mph)
knots (nautical miles per hour, symbol kt)
Mach, where Mach 1 is the speed of sound; Mach n is n times as fast.

Mach 1 = ~343 m/s = ~1235 km/h = ~768 mi/h (see the speed of sound for more detail)

speed of light in vacuum (symbol c) is one of the natural units

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.

The rate of change of speed with respect to time is termed acceleration.

1 Average speed
2 Cultural significance
3 See also
4 External links

Average speed

Speed as a physical property represents primarily instantaneous speed. In real life we often use average speed (denoted $\tilde{v}$ ), 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:

$\tilde{v} = \frac{\Delta l}{\Delta t}.$

Instantaneous speed defined as a function of time on interval $[t 0, t 1]$ gives average speed:

$\tilde{v} = \frac{\int_{t_0}^{t_1} v(t) \, dt}{\Delta t}$

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

$\tilde{v} = \frac{\Delta l}{\int_{l_0}^{l_1} \frac{1}{v(l)} \, dl}$

It is often intuitively expected that going half a distance with speed $v a$ and second half with speed $v b$ , produce total average speed $\tilde{v} = \frac{v_a + v_b}{2}$ . The correct value is $\tilde{v} = \frac{2}{\frac{1}{v_a} + \frac{1}{v_b}}$
(Note that the first is arithmetic mean while the second is harmonic mean).
Average speed can be derived also from speed distribution function (either in time or on distance):

$v \sim D_t\; \Rightarrow \; \tilde{v} = \int v D_t(v) \, dv$

$v \sim D_l\; \Rightarrow \; \tilde{v} = \frac{1}{\int \frac{D_l(v)}{v} \, dv}$

Cultural significance

Speed or swiftness of motion plays a significant role in human culture, see racing. It is complementary to grace, precision and strength, e.g. in dancing or martial arts. Animals symbolizing speed are the horse (PIE *ek'vos is etymologically derived from *ok'u- "swift"), birds, especially raptors such as the hawk, and cats, e.g. the lynx (see e.g. Flos Duellatorum). The swiftest land animal is the cheetah, reaching running speeds of up to 110km/h.