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:

$v\; =\; \backslash left|\backslash frac\; \{d\}\{t\}\backslash right|.$

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

Units of speed include:

• meters per second, (symbol ms^-1; 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 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

  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.

  Average speed

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

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

  Instantaneous speed defined as a function of time on interval $t\_1$ gives average speed:

  $|\backslash tilde\{v\}|\; =\; \backslash frac\{\backslash int\_\{t\_0\}^\{t\_1\}\; |v|(t)\; \backslash ,\; dt\}\{\backslash Delta\; t\}$

  while instantaneous speed defined as a function of distance (or length) on interval $l\_1$ gives average speed:

  $|\backslash tilde\{v\}|\; =\; \backslash frac\{\backslash Delta\; l\}\{\backslash int\_\{l\_0\}^\{l\_1\}\; \backslash frac\{1\}\; \backslash ,\; dl\}$

  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 $|\backslash tilde\{v\}|\; =\; \backslash frac\{2\}$. The correct value is $|\backslash tilde\{v\}|\; =\; \backslash frac\{2\}\{\backslash frac\{1\}\; +\; \backslash frac\{1\}\}$
  (Note that the first is a proper arithmetic mean while the second is a proper harmonic mean).