Speed

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For other uses, see Speed (disambiguation).

Speed is the rate of motion, or equivalently the rate of change of position, many times 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.

In mathematical notation, it is simply:

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

Units of speed include:

meters per second, (symbol m/s), the SI derived unit
kilometers 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 mph (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.

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

[edit] 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, but incorrect, that going half a distance with speed $v a$ and second half with speed $v b$ , produces 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 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):

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

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

[edit] See also

[edit] References

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This article has been tagged since December 2006.