# speed

Physical size
Surname speed
Formula symbol ${\ displaystyle v}$
Size and
unit system
unit dimension
SI m · s −1 L · T −1
cgs cm  ·  s −1 L · T −1
Planck c c

The speed is next to the site and the acceleration of the fundamental concepts of kinematics , a branch of mechanics . How fast and in what direction the speed describes a body , or a phenomenon (such as a wave crest ) in the course of time its location changed. A speed is indicated by its amount and the direction of movement; it is therefore a vector quantity . As a formula symbol is common after the Latin or English word for speed ( Latin velocitas , English velocity ). ${\ displaystyle {\ vec {v}}}$

Often the word speed only refers to its amount (formula symbol ), which, clearly speaking, reflects the current "speed" of the movement, as it is shown by the speedometer in a car, for example . indicates the distance a body covers within a certain period of time if the speed remains constant for a correspondingly long time. The unit used internationally is meters per second (m / s) , kilometers per hour (km / h) and - especially in sea and aviation - knots (kn) are also common . ${\ displaystyle v}$${\ displaystyle v}$

The highest possible speed with which the effect of a certain cause can spread spatially is the speed of light  . This upper limit also applies to any information transfer. Bodies that have a mass can only move at lower speeds than . ${\ displaystyle c}$${\ displaystyle c}$

A speed specification is always to be understood relative to a reference system . If a body is at rest in one frame of reference, then in another frame of reference, which moves with the speed of the first , it has the same speed in the opposite direction . ${\ displaystyle {\ vec {v}}}$${\ displaystyle - {\ vec {v}}}$

## Conceptual history and etymology

The exact formulation of the everyday concepts of speed and movement has been problematic since antiquity and throughout the Middle Ages (see, for example, " Achilles and the turtle " and the " arrow paradox "). The clarification in the physical sense comes from Galileo Galilei and marks the scientific breakthrough to modern physics at the beginning of the 17th century. Until then, only the average speed along a given finite distance had been precisely defined, and an increase in speed, such as in free fall, was imagined as the result of small jumps in the amount of speed. With Galileo, on the other hand, a constantly varying speed sweeps over a continuum of all intermediate values, which he did not understand as the average speed of a given piece of the route, but as the instantaneous speed at the respective point on the path. The exact version of this concept of speed using the border crossing to infinitely small distances was only given by Isaac Newton at the end of the 17th century . The two aspects of the magnitude and direction of the speed were initially only treated separately until they were merged into a single mathematical variable, the speed vector , in the 19th century .

The word speed goes back to Middle High German geswinde ('fast, vorschnell , impetuously, kühn'), Middle Low German geswint , geswine ('stark', meaning reinforced by the prefix ge ), Middle High German swinde , swint ('mighty, stark, violent, agile 'quick, angry, dangerous') back. Old high German occurrence is proven by names such as Amalswind, Swindbert, Swinda .

## definition

Path curve, position vector, chord and distance

The concept of speed (symbol ) at a certain point ( ) of the path curve is obtained approximately from the local change in the period : . ${\ displaystyle {\ vec {v}}}$${\ displaystyle A}$${\ displaystyle \ Delta {\ vec {r}}}$${\ displaystyle \ Delta t}$${\ displaystyle {\ vec {v}} _ {AB} = {\ frac {\ Delta {\ vec {r}}} {\ Delta t}}}$

Here, the chord of the section between the points and at which the body at the beginning or at the end of the period is. The vector has the direction of the chord . ${\ displaystyle \ Delta {\ vec {r}} = {\ vec {r}} _ {B} - {\ vec {r}} _ {A}}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle \ Delta t}$${\ displaystyle {\ vec {v}} _ {\ text {AB}}}$${\ displaystyle \ Delta {\ vec {r}}}$

The instantaneous speed at the point results from this speed if the point is allowed to come so close that the quotient tends towards a certain value, the limit value. At the same time, the time interval tends towards zero, which is written as. This process, called the border crossing , finds a mathematically exact basis in differential calculus . The current speed at the point is (in three equivalent notations) ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle {\ frac {\ Delta {\ vec {r}}} {\ Delta t}}}$${\ displaystyle {\ underset {\ Delta t \ rightarrow 0} {\ lim}}}$${\ displaystyle A}$

${\ displaystyle {\ vec {v}} = {\ underset {\ Delta t \ rightarrow 0} {\ lim}} {\ frac {\ Delta {\ vec {r}}} {\ Delta t}} = {\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm {d} t}} = {\ dot {\ vec {r}}}}$.
Distance in a movement with a curved trajectory.
Special case of linear motion.

Since the chord takes the direction of the tangent to the trajectory at the border crossing, this is also the direction of the instantaneous speed. ${\ displaystyle \ Delta {\ vec {r}}}$

The amount of the instantaneous speed (the "tempo" or the path speed ) is determined by the amount of the speed vector

${\ displaystyle v = \ left | {\ vec {v}} \ right | = \ left | {\ underset {\ Delta t \ rightarrow 0} {\ lim}} {\ frac {\ Delta {\ vec {r} }} {\ Delta t}} \ right | = \ left | {\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm {d} t}} \ right | = \ left | {\ dot {\ vec {r}}} \ right |}$

given, where is the absolute value of the position vector . The path speed is not the same as , for example, as can be seen from the circular motion . ${\ displaystyle \ left | {\ vec {r}} \ right | = r}$${\ displaystyle {\ vec {r}}}$${\ displaystyle \ left | {\ dot {r}} \ right |}$${\ displaystyle r = {\ text {const.}}, \ v \ neq 0, \ {\ dot {r}} = 0}$

The amount of the instantaneous speed can also be obtained as a scalar if, instead of the three-dimensional trajectory, only the distance (symbol ) along the trajectory is taken into account (see Fig.). It is the limit that the ratio of distance traveled and time needed assuming: . ${\ displaystyle s}$${\ displaystyle \ Delta s}$${\ displaystyle \ Delta t}$${\ displaystyle v = {\ underset {\ Delta t \ rightarrow 0} {\ lim}} {\ frac {\ Delta s} {\ Delta t}} = {\ frac {{\ text {d}} s} { {\ text {d}} t}} = {\ dot {s}}}$

## Average speed

If you divide the total distance covered by the total elapsed time to calculate the speed, you get the average speed as the result. The information about the change over time is lost. For example, if a car covers a distance of 100  km in one hour, it has an average speed of 100 km / h. It can actually have been driven at a constant speed of 100 km / h or a quarter of an hour at a speed of 200 km / h and three quarters of an hour at a speed of 66.7 km / h.

Note that the average speed is always the temporal represents mean value of the speed. If a car first drives at a constant speed of 50 km / h for half an hour and then for half an hour at a constant speed of 100 km / h, the average speed is 75 km / h. But if the car first drives a distance of 25 km at a constant speed of 50 km / h and then a distance of 25 km at a constant speed of 100 km / h, only half the time is required for the second movement segment (one Quarter of an hour). As a result, the average speed in this case is 66.7 km / h, although this may be contrary to intuition.

Another example of bodies with variable speed are celestial bodies, whose speeds vary on elliptical orbits around a central body. At Mercury the average speed is 47.36 km / s, but fluctuates between 39 and 59 km / s because of the noticeable eccentricity.

## Initial speed

If the speed of a body or a mass point at the beginning of a certain movement segment is of interest, it is also referred to as the initial speed (symbol usually v 0 ).

The initial speed is one of the initial conditions when solving the equations of motion in classical mechanics, for example for numerical simulations in celestial mechanics . It is an important parameter e.g. B. for the trajectory for vertical and oblique throw and for the range of firearms or missiles .

Examples:

## Top speed

The vertically downward weight force is equal to the vertically upward aerodynamic drag force . The forces cancel each other out so that the body does not experience any further acceleration. The top speed has been reached.${\ displaystyle {\ vec {G}} = {\ vec {F}} _ {\ mathrm {G}} = {\ vec {F}} _ {\ mathrm {g}}}$${\ displaystyle {\ vec {F}} _ {\ mathrm {d}}}$

The final speed (also: limit speed ) describes the speed that an object has reached at the end of its acceleration.

An object reaches its final speed when the braking forces have become so strong by increasing or decreasing the speed that an equilibrium of all forces involved develops. The acceleration when reaching the final speed is therefore zero.

The term is also used in technology . In the automotive sector, for example, one speaks of top speed or maximum speed when the vehicle cannot be accelerated further, limited by engine power and external circumstances.

## Relations to other physical quantities

From the time profile of the speed one can infer the distance traveled by about the time integrated : . In the simplest case, namely at constant speed, it becomes . ${\ displaystyle \ Delta s = \ int _ {t_ {1}} ^ {t_ {2}} v (t) \, \ mathrm {d} t}$${\ displaystyle \ Delta s = v \, \ Delta t}$

The first derivative of the velocity with respect to time is the acceleration : . ${\ displaystyle {\ vec {a}} (t) = {\ dot {\ vec {v}}} (t) = {\ ddot {\ vec {s}}} (t)}$

The second derivative of the velocity with respect to time gives the jerk one movement . ${\ displaystyle {\ vec {j}} (t) = {\ ddot {\ vec {v}}} (t) = {\ dot {\ vec {a}}} (t)}$

The momentum - in other words, the “swing” - of a body of mass is calculated from it , while the kinetic energy is given by . Strictly speaking, the last two equations only apply approximately for the so-called non-relativistic case, i.e. for speeds that are much smaller than the speed of light. ${\ displaystyle m}$${\ displaystyle {\ vec {p}} = m {\ vec {v}}}$${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} mv ^ {2} = {\ frac {p ^ {2}} {2m}}}$

## Measurement

The easiest way to determine the speed is to measure

• what time is required for a certain distance or
• which distance is covered in a given time interval.

In both cases, only an average speed is actually measured. However, if the path or time interval is chosen to be short enough or if the movement is approximately uniform, satisfactory accuracies can be achieved with both methods. An example of method 1 would be the measurement of the speed of light according to Hippolyte Fizeau . Method 2 is used, among other things, when speed values ​​are calculated from GPS data.

The speed of a vehicle can easily be determined with a speedometer. This actually measures the speed of the wheel, which is directly proportional to the speed.

However, practically any other speed-dependent effect can also be used for a measurement method, e.g. B. the Doppler effect in Doppler radar , the pulse in the ballistic pendulum or the dynamic pressure in the Prandtl probe .

## units

The SI unit of speed is meters per second (m / s). Another common unit of speed is kilometers per hour (km / h).

The term “ kilometers per hour ” is also used in everyday language . Since in physics such a combination of two units (here: “hour” and “kilometer”) is understood as a multiplication of these units, the term “kilometers per hour” is normally not used in the natural sciences.

The non-metric unit used in the United States and some other English-speaking countries is miles per hour ( mph ). The unit knot (kn) is also used in maritime and aviation . A knot is one nautical mile (sm) per hour. Vertical speeds in motorized aviation are often given in feet per minute (LFM from English linear feet per minute or just fpm from English feet per minute ).

The Mach number is almost only used in aviation. It does not indicate an absolute value, but rather the ratio of speed to the local speed of sound . The speed of sound is strongly temperature-dependent but not dependent on air pressure . The reason for using this number is that aerodynamic effects depend on it.

Conversion of common speed units:

Meters per second kilometers per hour Knots (= nautical miles per hour) miles per hour Speed ​​of Light
1 m / s 00 1 00 3.6 00 1,944 00 2.237 0 3.336 · 10 −9
1 km / h 00 0.2778 00 1 00 0.5400 00 0.6215 0 9.266 · 10 −10
1 kn 00 0.5144 00 1,852 00 1 00 1.151 0 1.716 · 10 −9
1 mph 00 0.4469 00 1.609 00 0.8688 00 1 0 1.491 · 10 −9
1 ${\ displaystyle c}$ 00 299792458 00 1.079 · 10 9 00 5.827 · 10 8 00 6.707 10 8 00 1

Note: The conversion factors in bold are exact , all others are rounded to the nearest four digits.

## Velocities and frame of reference

Passing aircraft with speed (red), radial speed (green) and tangential speed (blue)

Depending on the reference system or coordinate system used, different terms have become established:

A Cartesian coordinate system is often used in a homogeneous gravitational field . Velocities that are directed parallel to the gravitational acceleration are usually referred to as vertical velocities , while those that are orthogonal to this direction are referred to as horizontal velocities . ${\ displaystyle {\ vec {g}}}$

In the case of polar coordinates , the radial speed is the component of the speed vector in the direction of the position vector , i.e. along the connecting line between the moving object and the origin of the coordinates. The component perpendicular to this is called the peripheral speed . : Thus results . The vector product of the angular velocity and the position vector results in the peripheral speed : . ${\ displaystyle {\ vec {v}} _ {\ mathrm {r}}}$ ${\ displaystyle {\ vec {v}} _ {\ perp}}$${\ displaystyle {\ vec {v}} = {\ vec {v}} _ {\ perp} + {\ vec {v}} _ {\ mathrm {r}}}$${\ displaystyle {\ vec {v}} _ {\ perp} = {\ vec {\ omega}} \ times {\ vec {r}}}$

When moving on a circular path around the coordinate origin, but only in this case, the radial speed is zero and the peripheral speed is equal to the tangential speed, i.e. the path speed along the tangent to the path curve.

From the change in the distance to the origin of coordinates (radius) the radial velocity follows . ${\ displaystyle {\ vec {v}} _ {\ mathrm {r}} = {\ dot {r}} \, {\ frac {\ vec {r}} {| {\ vec {r}} |}} }$

Assuming that there is a generally valid reference system, the speeds measured in this system are called absolute speeds . Velocities that refer to a point that moves itself in this system are called relative velocities . Example: A tram travels at a speed of 50 km / h. A passenger moves in it at a relative speed (compared to the tram) of 5 km / h. Its absolute speed (as seen by the stationary observer on the road) is 55 km / h or 45 km / h, depending on whether he is moving in the direction of travel or against the direction of travel.

However, the principle of relativity says that there is no physical reason why one should single out a certain reference system and prefer it to other systems. All physical laws that apply in one inertial system also apply in every other. Which movements are viewed as "absolute" is therefore completely arbitrary. That is why the concept of absolute speed has been avoided since the special theory of relativity at the latest . Instead, all speeds are relative speeds. From this relativity principle, together with the invariance of the speed of light, it follows that speeds - as tacitly assumed in the above example - may not simply be added. Instead, the relativistic addition theorem applies to speeds. However, this is only noticeable at very high speeds.

## Speed ​​of numerous particles

Temperature dependence of the velocity distribution for nitrogen

If you consider a system made up of many particles, it is usually no longer sensible or even possible to specify a certain speed for each individual particle. Instead, one works with the velocity distribution, which indicates how often a certain range of velocities occurs in the ensemble of particles . In an ideal gas , for example, the Maxwell-Boltzmann distribution applies (see figure on the right): Most particles have a speed close to the most probable speed, which is indicated by the maximum of the Maxwell-Boltzmann distribution. Very small and very large velocities also occur, but are only accepted by very few particles. The position of the maximum depends on the temperature. The hotter the gas, the higher the most likely speed. More particles then reach high speeds. This shows that the temperature is a measure of the mean kinetic energy of the particles. However, even at low temperatures, very high speeds cannot be completely ruled out. The speed distribution can be used to explain many physical transport phenomena, such as B. diffusion in gases.

## Flow velocity of a fluid

The average flow velocity of a gas or a liquid resulting from the volume current through the flow cross-section : . However, the local flow velocities can differ greatly from one another. For example, the speed is greatest in the middle of an ideal pipe and drops to zero due to the friction towards the wall. One must therefore understand the flow of a medium as a vector field . If the velocity vectors are constant over time, one speaks of a steady flow . In contrast, if the velocities behave chaotically, it is a turbulent flow . The Reynolds number helps to characterize the flow behavior, as it relates the flow velocity to the dimensions of the flowed body and the viscosity of the fluid. ${\ displaystyle v _ {\ mathrm {A}}}$${\ displaystyle Q = {\ tfrac {\ mathrm {d} V} {\ mathrm {d} t}}}$${\ displaystyle A}$${\ displaystyle v _ {\ mathrm {A}} = {\ frac {Q} {A}}}$

Mathematically, the behavior of the speeds is modeled by the Navier-Stokes equations , which, as differential equations, relate the speed vectors to internal and external forces. They have a similar meaning for the motion of a fluid as the basic equation of mechanics for mass points and rigid bodies.

## Speed ​​of waves

The red point moves at phase speed and the green points at group speed.

The complex movement of waves makes it necessary to use different concepts of speed. (In particular, the word speed of propagation can mean various things.)

• The speed of deflection of mechanical waves is called the speed . The best-known example is the oscillation speed of air particles in a sound wave.
• The speed at which a point in a certain phase moves forward is called the phase speed . The following applies: . Here are the wavelength, the period, the angular frequency and the circular wave number. The speed at which the crests of the waves move in the ocean is a typical example of a phase speed.${\ displaystyle v _ {\ mathrm {p}} = {\ frac {\ lambda} {T}} = {\ frac {\ omega} {k}}}$${\ displaystyle \ lambda}$${\ displaystyle T}$${\ displaystyle \ omega}$${\ displaystyle k}$
• The speed with which a whole wave packet moves is group velocity called: .${\ displaystyle v _ {\ mathrm {g}} = {\ frac {\ partial \ mathbf {\ omega}} {\ partial \ mathbf {k}}}}$

Phase and group velocities only rarely coincide (e.g. propagation of light in a vacuum). Usually they are different. A vivid extreme example is the movement of snakes: If the snake is understood as a wave, the speed of its advancement is a group speed. However, the phase speed is zero when snaking, because the places where the body of the snake curves to the right or left are determined by the ground and do not move across the ground.

As a rule, the phase velocity of a physical wave depends on the frequency or the circular wave number. This effect is known as dispersion . Among other things, it is responsible for the fact that light of different wavelengths is refracted to different degrees by a prism .

## theory of relativity

From the laws of classical physics it follows for speeds, among other things:

• The measured values ​​for lengths and times are independent of the state of motion (and thus the speed) of the observer . In particular, all observers agree on whether two events take place at the same time or not.
• The Galileo transformation applies when the reference system is changed . This means that the speeds of movements that are superimposed can be added vectorially.
• There is no theoretical upper limit to the speed of movement.
• It is not required by the laws of classical physics, but before Einstein it was generally assumed that there was a universal reference system, the “ ether ”, for all speeds . If that were the case, the speed of propagation of electromagnetic waves would have to depend on the state of motion of the receiver.

The latter dependency could not be demonstrated with the Michelson-Morley experiment . Einstein postulated that the principle of relativity, which was already known from classical mechanics, must also be applied to all other phenomena in physics, especially the propagation of light, and that the speed of light is independent of the state of motion of the transmitter. From this he concluded that the above statements of classical mechanics must be modified. In detail this means:

• The measured values ​​for lengths and times depend on the state of motion (and thus the speed) of the observer (see time dilation and length contraction ). Simultaneity is also relative .
• The Lorentz transformation applies when the reference system is changed . This means that speeds of movements that are superimposed cannot simply be added vectorially.
• Bodies can only move at speeds that are less than the speed of light. Information cannot be transmitted faster than light either.
• There is no “ether”.

The effects that result from the special theory of relativity, however, only become noticeable at very high speeds. The Lorentz factor , which is decisive for time dilation and length contraction, only results in a deviation of more than one percent for speeds of . Consequently, classical mechanics represent an extremely precise approximation even for the fastest spacecraft ever built. ${\ displaystyle v> 4 {,} 2 \ cdot 10 ^ {7} \, \ mathrm {\ tfrac {m} {s}}}$