In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the

^{−1} or m/s), the

Section 8–2

magnitude
Magnitude may refer to:
Mathematics
* Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order o ...

of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...

travelled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero. Speed is not the same as velocity.
Speed has the dimensions of distance divided by time. The SI unit of speed is the metre per second
The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre ...

(m/s), but the most common unit of speed in everyday usage is the kilometre per hour (km/h) or, in the US and the UK, miles per hour
Miles per hour (mph, m.p.h., MPH, or mi/h) is a British imperial and United States customary unit of speed expressing the number of miles travelled in one hour. It is used in the United Kingdom, the United States, and a number of smaller cou ...

(mph). For air and marine travel, the knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

is commonly used.
The fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum ''c'' = metres per second (approximately or ). Matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...

cannot quite reach the speed of light, as this would require an infinite amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed.
Definition

Historical definition

Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time.Hewitt (2006), p. 42 In equation form, that is :$v\; =\; \backslash frac,$ where $v$ is speed, $d$ is distance, and $t$ is time. A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion often have variations in speed (a car might travel along a street at 50 km/h, slow to 0 km/h, and then reach 30 km/h).Instantaneous speed

Speed at some instant, or assumed constant during a very short period of time, is called ''instantaneous speed''. By looking at aspeedometer
A speedometer or speed meter is a gauge that measures and displays the instantaneous speed of a vehicle. Now universally fitted to motor vehicles, they started to be available as options in the early 20th century, and as standard equipment ...

, one can read the instantaneous speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance (25 km). If it continued for only one minute, it would cover about 833 m.
In mathematical terms, the instantaneous speed $v$ is defined as the magnitude of the instantaneous velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

$\backslash boldsymbol$, that is, the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of the position $\backslash boldsymbol$ with respect to time:
:$v\; =\; \backslash left,\; \backslash boldsymbol\; v\backslash \; =\; \backslash left,\; \backslash dot\; \backslash \; =\; \backslash left,\; \backslash frac\backslash \backslash ,.$
If $s$ is the length of the path (also known as the distance) travelled until time $t$, the speed equals the time derivative of $s$:
:$v\; =\; \backslash frac.$
In the special case where the velocity is constant (that is, constant speed in a straight line), this can be simplified to $v=s/t$. The average speed over a finite time interval is the total distance travelled divided by the time duration.
Average speed

Different from instantaneous speed, ''average speed'' is defined as the total distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, the average speed is 80 kilometres per hour. Likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres (km) is divided by a time in hours (h), the result is in kilometres per hour (km/h). Average speed does not describe the speed variations that may have taken place during shorter time intervals (as it is the entire distance covered divided by the total time of travel), and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to :$d\; =\; \backslash boldsymbolt\backslash ,.$ Using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Expressed in graphical language, theslope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...

of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval covered by the chord. Average speed of an object is
Vav = s÷t
Difference between speed and velocity

Speed denotes only how fast an object is moving, whereasvelocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

describes both how fast and in which direction the object is moving. If a car is said to travel at 60 km/h, its ''speed'' has been specified. However, if the car is said to move at 60 km/h to the north, its ''velocity'' has now been specified.
The big difference can be discerned when considering movement around a circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

. When something moves in a circular path and returns to its starting point, its average ''velocity'' is zero, but its average ''speed'' is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average ''velocity'' is calculated by considering only the displacement between the starting and end points, whereas the average ''speed'' considers only the total distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...

travelled.
Tangential speed

Linear speed is the distance travelled per unit of time, while tangential speed (or tangential velocity) is the linear speed of something moving along a circular path.Hewitt (2006), p. 131 A point on the outside edge of amerry-go-round
A carousel or carrousel (mainly North American English), merry-go-round (international), roundabout (British English), or hurdy-gurdy (an old term in Australian English, in SA) is a type of amusement ride consisting of a rotating circular p ...

or turntable
A phonograph, in its later forms also called a gramophone (as a trademark since 1887, as a generic name in the UK since 1910) or since the 1940s called a record player, or more recently a turntable, is a device for the mechanical and analogu ...

travels a greater distance in one complete rotation than a point nearer the center. Travelling a greater distance in the same time means a greater speed, and so linear speed is greater on the outer edge of a rotating object than it is closer to the axis. This speed along a circular path is known as ''tangential speed'' because the direction of motion is tangent to the circumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and both use units of m/s, km/h, and others.
Rotational speed
Rotational frequency (also known as rotational speed or rate of rotation) of an object rotating around an axis is the frequency of rotation of the object. Its unit is revolution per minute (rpm), cycle per second (cps), etc.
The symbol fo ...

(or ''angular speed'') involves the number of revolutions per unit of time. All parts of a rigid merry-go-round or turntable turn about the axis of rotation in the same amount of time. Thus, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time. It is common to express rotational rates in revolutions per minute (RPM) or in terms of the number of "radians" turned in a unit of time. There are little more than 6 radians in a full rotation (2 radians exactly). When a direction is assigned to rotational speed, it is known as rotational velocity or angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

. Rotational velocity is a vector whose magnitude is the rotational speed.
Tangential speed and rotational speed are related: the greater the RPMs, the larger the speed in metres per second. Tangential speed is directly proportional to rotational speed at any fixed distance from the axis of rotation. However, tangential speed, unlike rotational speed, depends on radial distance (the distance from the axis). For a platform rotating with a fixed rotational speed, the tangential speed in the centre is zero. Towards the edge of the platform the tangential speed increases proportional to the distance from the axis. In equation form:
:$v\; \backslash propto\; \backslash !\backslash ,\; r\; \backslash omega\backslash ,,$
where ''v'' is tangential speed and ω (Greek letter omega
Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. Th ...

) is rotational speed. One moves faster if the rate of rotation increases (a larger value for ω), and one also moves faster if movement farther from the axis occurs (a larger value for ''r''). Move twice as far from the rotational axis at the centre and you move twice as fast. Move out three times as far, and you have three times as much tangential speed. In any kind of rotating system, tangential speed depends on how far you are from the axis of rotation.
When proper units are used for tangential speed ''v'', rotational speed ω, and radial distance ''r'', the direct proportion of ''v'' to both ''r'' and ω becomes the exact equation
:$v\; =\; r\backslash omega\backslash ,.$
Thus, tangential speed will be directly proportional to ''r'' when all parts of a system simultaneously have the same ω, as for a wheel, disk, or rigid wand.
Units

Units of speed include: * metres per second (symbol m sSI derived unit
SI derived units are units of measurement derived from the
seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate ...

;
* kilometres per hour (symbol km/h);
*miles per hour
Miles per hour (mph, m.p.h., MPH, or mi/h) is a British imperial and United States customary unit of speed expressing the number of miles travelled in one hour. It is used in the United Kingdom, the United States, and a number of smaller cou ...

(symbol mi/h or mph);
* knots ( nautical miles per hour, symbol kn or kt);
*feet per second The foot per second (plural feet per second) is a unit of both speed (scalar) and velocity (vector quantity, which includes direction). It expresses the distance in feet (ft) traveled or displaced, divided by the time in seconds (s). The corresp ...

(symbol fps or ft/s);
* Mach number ( dimensionless), speed divided by the speed of sound;
*in natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ...

(dimensionless), speed divided by the speed of light in vacuum (symbol ''c'' = ).
Examples of different speeds

Psychology

According to Jean Piaget, the intuition for the notion of speed in humans precedes that of duration, and is based on the notion of outdistancing. Piaget studied this subject inspired by a question asked to him in 1928 by Albert Einstein: "In what order do children acquire the concepts of time and speed?" Children's early concept of speed is based on "overtaking", taking only temporal and spatial orders into consideration, specifically: "A moving object is judged to be more rapid than another when at a given moment the first object is behind and a moment or so later ahead of the other object."See also

References

* Richard P. Feynman, Robert B. Leighton, Matthew Sands.The Feynman Lectures on Physics
''The Feynman Lectures on Physics'' is a physics textbook based on some lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the Cali ...

, Volume ISection 8–2

Addison-Wesley
Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles throug ...

, Reading, Massachusetts (1963). .
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Physical quantities
Temporal rates
Velocity