Relative speed

The expression relative speed is commonly used, to express the difference of the Speed S of two mobiles or the variation in the Temps of the distance between two mobiles.
It is also employed to express variations compared compared to the time of quantities other than of the distances: relative Speed of growth (of the weight, size etc).
Aussi simple that it can appear with the first access, this concept, according to the context where it is used, requires precise definitions of the various concrete objects (material) or theoretical (mathematics) which it implements. Of the motorist who is made exceed on a highway by a vehicle of which he wants to estimate speed, to the scientist seeking the method of the accosting of two space engines most economic in energy, while passing through the navigator checking that he is not on the way of collision with another ship, one imagines easily that the methods of measurements and calculations will be different.
The appreciation of the distances, the rates of travel, the complexity of the trajectories, the precision of measurements and the results, depend on the scope of application.

Limits of the subject of the article on relative speed

It is a speed applying to the distances covered in space

If speed expresses in a general way the variation of a quantity (other that time, like the temperature, the pressure, weight etc) compared to one duration (variation of time), which follows relates to only the fields of the Cinématique and mechanics: Mechanical Newtonian or Mechanical relativist. By speed one will thus understand distance covered per unit of time. Relative speed being a speed, it could be defined, according to the needs, by only one scalar size (20 km/h, 1 turn per minute) or by several sizes making it possible to specify the characteristics of them, like the direction and the direction. (See in the article on the Speed: Vector speed, angular Velocity, areal speed, instantaneous speed, mean velocity, curvilinear speed.)

Relativity

The examples which follow do not aim to clarify the theories of relativistic physics (restricted Relativité, General relativity). However the interest that they present will be approached, when traditional mechanics reaches its limits of validity (in coherence or precision).

Vocabulary

It will be able to prove to be useful to consult the article devoteds to in the certain scientific terms like reference frame galiléen or of trade, in particular for the examples concerning the sea Transport.

Opening remarks on the concept relative speed

Relative speed qualifies the variation of distance between two mobiles per unit of time, or the speed of a mobile observed since another mobile. The expression relative speed is sometimes replaced by other equivalent terms:

• apparent speed, for the apparent Wind in sea transport, different the real wind (perceived by a ship with the stop, damping),
• speed indicated, calibrated airspeed, in aeronautical navigation (see Speeds (aerodynamics)), different the true speed,
• speed of approach, but also relative speed in space navigation, for accostings of space engines.

In all the cases relative speed is a speed observed in an even mobile reference frame him in another reference frame (absolute or not). The concept relative speed has direction only between two animated entities a speed in a reference frame which does not depend on them, but where they evolve/move.

Opening remarks on the reference frames

The reference frame is a system which makes it possible to observe and note (to locate) the successive positions of a mobile in space and time, this system being invariant throughout observation. A reference frame thus includes/understands at least an axis of space (a space dimension) and an axis of times (a dimension temporal, a priori independent of space) which make it possible an observer to locate an object (fixed or mobile). In mechanics, the reference frames have scientifically adapted definitions. (See Reference frame galiléen.) It is remarkable that the definition of the reference frames is important to establish the mathematical formulas which make it possible to pass from a reference frame to another by preserving the laws of physics, according to each approach of physics (Lois of Newton, restricted Relativité, General relativity, quantum Mécanique).
In what follows, the word reference frame will be generally used to indicate a space frame of reference (on a dimension, a plan (2 dimensions), a volume (3 dimensions)) and a stop watch.

Definitions relative speed

The relative Speed of a M1 mobile compared to a mobile m2 is the rate of travel of M1 observed since m2.
Or: the relative speed of two mobiles is speed where these two mobiles approach or move away one from the other.
Or: the relative speed of two mobiles is the difference speeds of these mobiles.
These preceding definitions, apparently simple, relative speed are not equivalent and under hear the existence nonrelative, absolute speeds.
They are not either rigorous, even false, with respect to the principles of the Newtonian Mécanique or the Mécanique relativist.
A more scientific definition could be: the relative speed of a mobile is its speed in an even mobile reference frame him in another reference frame (what is redundant in a mechanics not considering that there does not exist absolute reference mark). The scientific definition is: the relative speed of a mobile is its speed in a reference frame. It has the advantage of being simple and the disadvantage of not making the difference between a speed and a relative speed, which closes the subject prematurely.
The definitions relative speed being now outlined, it is interesting to specify them by appreciating their use in some examples.

Concrete examples of use relative speeds in a current environment

By environment running it is necessary to include/understand the daily human environment where the results of calculations are of a sufficient precision, taking into account the approximation of the observations. (Newtonian Mechanics.)

Relative speeds of terrestrial vehicles

Two cars circulate on a Route two-track at constant speeds of 50 km/h and 60 km/h. They roll in their corridor of circulation in opposite directions and thus will cross. That the road is rectilinear or sinuous, that it goes up and that it goes down, and although the ground is round, figure 1 of diagram 1 is a representation of their situation in a Cartesian reference mark whose main axis OX corresponds to the line (white or yellow) separating the two lanes, the other axis, perpendicular, making it possible to represent them in a plan (plane diagram).
One can thus define three reference frames:

• a reference frame road, Rr, whose main axis is OX, where the followed road is developed in straight line,
• a reference frame A1 car, RA1, whose main axis is O' X, O' corresponding to the position of A1 on OX, moving away from O at the speed V1
• a reference frame (not represented on the diagram) RA2 for a2 car with an axis of O" origin; directed towards O.

These definitions of reference mark being made, two cars being comparable at the mobile points O' and O" , one can calculate following relative speeds:

• the relative Flight Path Vector of A2 compared to A1, is the speed of A2 in RA1, is $\backslash \; vec\; \{Vr\}\; =\; \backslash \; vec\; \{V\_2\}\; -\; \backslash \; vec\; \{V\_1\}$
• the relative Flight Path Vector of A1 compared to A2 is $\backslash \; vec\; \{Vr\text{'}\}\; =\; \backslash \; vec\; \{V\_1\}\; -\; \backslash \; vec\; \{V\_2\}\; =\; -\; \backslash \; vec\; \{Vr\}$
• A1 and A2 approaches one the other at a speed of $\backslash \; left|\; V\_1\; -\; (-\; V\_2)\; \backslash \; right|\; =\; \backslash \; left|\; V\_2\; -\; (-\; V\_1)\; \backslash \; right|$, is 50 (- 60) = 60 (- 50) = 110 km/h.

Relative speeds are here, such as the reference frames were defined, the result of the difference speeds which one will describe as “absolute”. Indeed, on the axis OX, O being an observer of the movements of O' (automobile A1) and O" (A2 car), the relative speed of the two mobiles is well also the variation in the time of the curvilinear distance (on the Bitume of the road) which separates them. However, it is an approximation, the two mobiles not following strictly the same course, this one having already been compared to that of the white line.
Let us suppose that we take another reference frame, based on the road map where the two cars circulate. We could find ourselves, according to the sinuosity of the course, at one moment T, in the situation of figure 2 of diagram 1. Relative speeds could, at the moment T, to be the subject of the same operations of subtraction (or addition) of vectors. But they would not have any utility, speeds of the two mobiles in this variable reference frame in time in standard and direction. Without going until a mountainous spherical reference frame, where one would try to make the difference in their absolute velocities to obtain their relative speeds!
On the other hand, if figure 2 of diagram 1 represents two mobiles in uniform rectilinear translatory movement in reference mark OX-OY, then $\backslash \; vec\; \{Vr\}$ represents well the relative speed of A2 compared to A1, $\backslash \; vec\; \{Vx\text{'}\}$ and $\backslash \; vec\; \{Vy\text{'}\}$ its components in the reference mark related to A1 (of which one of the axes, O' x', represents its own road).
In other words, the additions or subtractions of Flight Path Vectors between reference mark have direction only in referential galiléens.

Relative speeds in sea transport

Relative winds and relative speeds

In Navigation, that it is air or maritime, the roads are theoretical ways, not placing at the disposal of the navigator neither of the white lines nor of the Délinéateur S. Diagram 2 represents a sailing sailing ship on a sea moving on the bottom (ground) according to marine Running or a Courant of tide. It steers a course (course compass), compared to North, one of the axes of a terrestrial reference mark. The axis of the ship is directed according to this direction, and the ship follows an apparent road according to this cap.
We are in the presence of one only following mobile of the roads differing according to the reference frame: terrestrial reference frame (bottom), reference frame surfaces (sea). Measurements that the navigator (the observer) can carry out, are referred, are relative to its own reference frame:

• speed connects on water according to its course, indicated by its log,
• apparent speed of the wind in intensity and orientation (Vent apparent), indicated by its Anémomètre and its wind vane,

Other data are provided to him in a terrestrial reference frame:

• vectors running hour in hour.

If the navigator wants to know the real wind (wind in a terrestrial reference mark) it is necessary for him to estimate his speed on the bottom, therefore the speed of drive of its reference frame in the terrestrial reference frame. It will make simple additions or subtractions of vectors to determine its true road.
If he wants to correct (to inflect) his road according to foreseeable modifications wind or current velocities (weather), he will make forecasts hour per hour (currents being indicated hour per hour), by considering that between two positions separated by one hour, all the relative movements are uniform (Flight Path Vectors).
It is to be noticed that the reference frame of the mobile is here based on a reference mark

• moving linear and uniform (in a short time interval) compared to the terrestrial reference frame materialized by a Sea chart,
• centered on its apparent road.

Relative speeds on the way of collision

Two ships N1 and N2 follow roads surfaces some as been reproduced on diagram 3. N1 travels in the West with 5 nodes (Thousand sailor per hour), N2 with 10 nodes towards North (on diagram 3, as on the charts, North is in top).

Whatever the vector running, supposed the same one on all the zone, an observer on N2, observing that the Relèvement) of N1 is constant and noting visually that the boat approaches (by its size connects) in deduced logically that it will enter in collision with N1 (even observation since N1 with respect to N2). In the N2 reference frame, (N2 being the origin of a reference mark whose axis is its road, the other being perpendicular) the relative speed, or connects, of N1 is directed towards the N2 origin.

Their relative Flight Path Vector is equal to the vectorial difference of their Flight Path Vectors on the surface (or their Flight Path Vectors melts).

A N4 ship is with the anchor; its speed bottom is null but its speed surface equal and is opposed to the current of 5 nodes, coming from the West. N3 travels in North with 10 nodes. N3 notes that it is collision with N4 on the way because it perceives a relative speed of N4 directed towards him in its référentiel.
N3 thus makes the same observations as N2. If N3 does not know the existence of the current and does not know that N4 is with damping (anchored on the bottom); it will have the same perception of apparent road, relative speed, in its reference frame, as N2 in his.
The road of collision is a road on the bottom between N3 and N4, it is the road " vraie" , followed by N3 because N4 is fixed in a reference mark related to the bottom.
The road of N1 collision and N2 is a road observed in the reference frames related to each ship. It is only theoretical in a reference mark determined by a fixed observer on sea surface, i.e. an observer located on a ship at the stop on water, drifting like the current. Any observer fixed compared to sea, (like N1 and N2 if they give for origin of their reference frame a their opinion at a given moment), will see N1 and N2 to follow two convergent roads in a point where they will be present at the same moment.
In the example of diagram 3, N1 and N2 follow roads, on sea surface, perpendiculars (South-North and East-West) crossing at the time of the collision.

N1 and N2 follow, either of the roads leading them to the collision (reference frame surfaces), or a road of collision (reference frames related to the mobiles).
According to the relative movement (speed of drive due to the current) of the reference frame surfaces compared to a terrestrial reference frame (bottom), the two ships will also follow two different roads (according to their speeds compared to the bottom) carrying out them, at the same moment, whatever the current, with a collision.

How to establish a reference frame related to a mobile

The reference frames related to the mobiles are given starting from a reference frame which is common for them. To appreciate the fact that they are on the way of collision the two mobiles will locate themselves compared to:

• an angle (layer) compared to their Flight Path Vector surfaces (on water), and they suppose that they follow course constant on water and that sea surface is reference mark reliable (not absolute, but almost, the currents which they undergo being identical),
• or an angle compared to North (Relèvement), therefore compared to the bottom, which they also regard as a fixed reference frame (absolute), or centers it given by a Gyroscope. What returns to the same axis if the currents are fixed during the observation.

What is equivalent to say that the observer of a ship considers (that time is uniform, the question not being posed really) only the space which it defines (between its ship and the ship which approaches) is fixed in dimensions (the distances between two unspecified points of this space do not vary according to time). In both cases he considers that the current of surface, if there exists, follows a constant direction compared to the bottom, that the various drifts are constant. The observer, in the preceding examples, thus establishes a reference frame where even the origin represents to him (or its position at a given moment) and an axis directed compared to another reference frame, in fact its Flight Path Vector in another reference frame. If the observer does not have any means of knowing its Flight Path Vector in a reference frame where it evolves/moves (not of compass, no velocity measurement, therefore no external reference mark), the presence of an object will enable him to determine an axis between him and the object. If it can measure (to estimate) a distance, the variation in the time (its axis of times) of this distance will enable him to calculate their relative speed. With long distance they are a priori on a road of collision. It is only when the other object takes a measurable dimension which it can raise the doubt. (See Parallax)

Complex relative speeds: or not uniform nonlinear movements

The concept relative speed becomes more complex as soon as the mobiles do not move any more in reference marks galiléens (rectilinear motions at constant speed). While remaining in the fields where it is not necessary to utilize the theories of relativity, some examples make it possible to judge, in traditional mechanics, of the importance of the definition of the reference frames.

Accelerated linear movements

A simple example is that of the passenger of a car in acceleration or with braking. In this case the two mobiles are the car and the passenger, the reference mark are the road (in straight line), the reference mark related to the car and the reference mark related to the passenger. Colinearity of the movements, or at least implicit colinearity of the trajectories (parallel and close, therefore confused), since in a reduced space they easily make it possible to admit that locally the road constitutes a reference frame galiléen and that the main axis is common, make it possible to continue to admit that their relative instantaneous speeds are the difference in their absolute velocities;

If one names speed of drive speed that a mobile (the driver) would have in the accelerated reference frame (the car), which is in this case the absolute velocity of the car (in a reference mark not accelerated, or galiléen), one can always write that the Flight Path Vectors in the accelerated reference frame are such as:

$\backslash \; vec\; \{v\}\; \_\; \{relative\}\; =\; \backslash \; vec\; \{v\}\; \_\; \{absolute\}\; -\; \backslash \; vec\; \{v\}\; \_\; \{drive\}$,

like for their accelerations:

$\backslash \; vec\; \{has\}\; \_\; \{relative\}\; =\; \backslash \; vec\; \{has\}\; \_\; \{absolute\}\; -\; \backslash \; vec\; \{has\}\; \_\; \{drive\}$

If one now considers two mobiles located in a reference frame " absolu" or galiléen, following trajectories not colinéaires, of which one (M1) is animated a variable speed, accelerated, as on diagram 4, one can note that:

• the mobile m2 (at constant speed in the “absolute” reference frame) does not follow a rectilinear trajectory in the noninertial Référentiel of M1. Its relative speed varies (its Flight Path Vector varies in standard and direction); it is thus equipped, for M1, of an apparent relative acceleration whereas in “absolute” it does not have any;
• the M1 mobile, seen per m2, also seems to follow a curved trajectory;
• the two trajectories are symmetrical compared to the origin of the relative reference marks;

The layouts could correspond to the roads right and divergent followed by two cars of which one accelerates (M1) and the other maintains a speed constant (m2). The relative reference marks, confused here on diagram 4, would be then:

• OX: the followed road, in direction in direction, by each one of the vehicles,
• OY: 90° on the left of the followed road.

If instantaneous relative speeds are still equal to the difference in the instantaneous absolute velocities, it is not obviously any more the case for accelerations, since the relative trajectories are curvilinear, therefore that relative speeds change direction.

Relative movements in rotation and translation

Diagram 5 visualizes the courses of the same mobile M, in uniform linear trajectory (at constant speed) in a reference mark of observation, and its trajectories seen in a reference mark related to the disc in uniform rotation. (For the mathematical formulas, to see the article devoteds to with the Force of Coriolis and the Acceleration of Coriolis.)
On this diagram 5, only the trajectories are represented. Speeds can think, they are at every moment tangent with the trajectories.
Certain Flight Path Vectors are detailed on diagram 6. Diagram 5 described:

• a disc centered out of O, turning at number of constant revolutions (direction of the arrow, is trigonometrical or anti-clockwise) around O,
• a mobile M crossing the disc at uniform speed.

This mobile is for example an object slipping without friction on the disc (like a wet cake of soap), or an object rolling to the top of the disc in a gutter. M is thus an animated point (in a reference frame galiléen) a constant rectilinear speed. Its movement is independent of the movement of the rotation movement of the disc. The curves describe the relative trajectories of the point M, seen by any fixed point on the disc:

• AOC (in red) is the apparent trajectory of the point M when the disc makes 1/4 of turn while M traverses a diameter exactly of it;
• AOA (in blue) if the disc makes 1/2 turn while M crosses it;
• AOB (in green) if the disc makes 1 turn.

This diagram makes it possible to visualize the construction geometrical of the relative trajectories, where the points are raised in 1/16e of turn and 1/16e of diameter. To study relative speeds more closely diagram 6 details the case where the disc makes 1/2 turn while M traverses a diameter. In an absolute reference frame galiléen, R, that of the drawing, when has is in A1, M are in M1 and the reference frame R' de A1 is A1X', A1Y'. Projected in R, the relative Flight Path Vector of M1, in black, is the vectorial difference the absolute velocity of M1 (in uniform translation in R) and its speed of drive, i.e. the speed which M1 would have if it were fixed in R'. In other words, A1 sees to move away M1 with a speed made up of a translatory movement (real in the absolute) and of a rotation movement (apparent since M1 does not turn in the absolute). If has set a reference frame of projection, where it raises the successive positions of M (here AX, AY), it will locate at this moment that M is in Me 1, and equipped a relative speed Vrelative (Me 1/AX, AY), in blue.

• the relative speed of M, seen by has, is not equal to the difference in their absolute velocities because the reference frame of has is not galiléen.

It goes differently for the relative speed of, saw from there per M, whose trajectory in R' is traced in red on its projection in its relative reference mark AX, AY.

• As M moves at constant speed in R, and thus that its relative reference mark R' is galiléen, the relative speed of the other mobiles is equal to the difference in the absolute velocities (vectors in green):

$\backslash \; vec\; \{v\}\; \_\; \{relativedeA/R\text{'}\; deM\}\; =\; \backslash \; vec\; \{v\}\; \_\; \{absoluedeA\}\; -\; \backslash \; vec\; \{v\}\; \_\; \{absoluedeM\}$

As the acceleration of drive is null, and that the relative reference mark R' related to M is not in rotation (null Accélération of Coriolis):

$\backslash \; vec\; \{has\}\; \_\; \{relativedeA/R\text{'}\; deM\}\; =\; \backslash \; vec\; \{has\}\; \_\; \{absoluedeA\}$

• One can also notice that the relative trajectories are not symmetrical, which lets suppose that variation in the time relative speeds, therefore relative accelerations, are not either similar according to whether one observes a mobile since a Référentiel galiléen or a noninertial Référentiel. The mobile M, seen of a reference frame related to has, in translation and especially in rotation compared to the absolute reference frame, has a relative acceleration made up of an acceleration of drive (due to the movement of swing drive of the relative reference frame compared to the absolute reference frame) and of a complementary acceleration, or Accélération of Coriolis.

$\backslash \; vec\; \{v\}\; \_\; \{relativedeM/R\text{'}\; deA\}\; =\; \backslash \; vec\; \{v\}\; \_\; \{absoluedeM\}\; -\; \backslash \; vec\; \{v\}\; \_\; \{entrainementdeM\}$

The apparent deformation of the trajectory of M, which is linear in the absolute reference frame, thus comes owing to the fact that its speed of drive is not constant. What thus equips its relative speed of an acceleration not colinéaire:

$\backslash \; vec\; \{has\}\; \_\; \{relativedeM/R\text{'}\; deA\}\; =\; \backslash \; vec\; \{has\}\; \_\; \{entrainementdeM\}\; -\; \backslash \; vec\; \{has\}\; \_\; \{Coriolis\}$
(Here, M moving at constant absolute velocity, absolute acceleration is null.)

In the general case, the law of composition of the movements gives, for speeds and relative accelerations of a mobile observed since a reference frame not galiléen:

$\backslash \; vec\; \{v\}\; \_\; \{relative\}\; =\; \backslash \; vec\; \{v\}\; \_\; \{absolute\}\; -\; \backslash \; vec\; \{v\}\; \_\; \{drive\}$

$\backslash \; vec\; \{has\}\; \_\; \{relative\}\; =\; \backslash \; vec\; \{has\}\; \_\; \{absolute\}\; -\; \backslash \; vec\; \{has\}\; \_\; \{drive\}\; -\; \backslash \; vec\; \{has\}\; \_\; \{Coriolis\}$

This law of composition of the movements is resulting from formulas of changes of reference mark.

The most known observations, making it possible to check it, are:

• the Pendulum of Foucault,

• the weather phenomena which relate to the rotation of the winds,
• the Déviation towards the East of the fall of a body.

Formulas of change of reference mark

Change of reference mark in traditional mechanics

From the examples above, one can consider that the only definition the relative speed which remains indisputable, if one admits a universal time, is thus:

• the relative speed of a M1 mobile compared to a mobile m2 is the variation in the time of the vector position of M1 in the reference frame of m2.

It is starting from this definition that were elaborate the laws of Composition of the movements making it possible to calculate the Flight Path Vectors, derived from the vector position compared to time and the accelerations, derived from the Flight Path Vector compared to time.

One can note on diagram 7, that relative speeds depend, not only of the movement of the origin of the relative reference mark (has on diagram 7), but of the orientation of this relative reference mark compared to the reference mark of reference.

• relative speeds between two mobiles are related to the concept of Orientation.
• the orientation of the relative reference frame can induce a Rotation.

The notion speed of drive is made complicated of the fact even of rotation: it is not the rotation of the point origin of the relative reference frame which has importance, but the rotation of the relative reference frame compared to the reference mark of reference.

It is that quite naturally, because of the felt physical one (acceleration of the Pesanteur, centrifugal forces), we tend to take for reference frame a reference mark based on our movement and what the influence directly, thus determining local axes:

• the local vertical (terrestrial attraction),
• the terrestrial magnetic poles (Northern, Southern).

What complicates us the mathematical expression of the daily phenomena.

The mathematical formulas of change of reference mark are however simple:

For the point P, all derivations (and the representations of the vectors) being made in R:

• its position in the reference mark R being represented by the vector $\backslash \; vec\; \{COp\}$,

• its position in the R' reference mark being represented by the vector $\backslash \; vec\; \{AP\}$,
• the speed of the point P in R of origin O is: $\backslash \; frac\; \{D\; \backslash \; vec\; \{COp\}\}\; \{dt\}$, derived from the vector position compared to time,
• the speed of the point P in R' of origin has is: $\backslash \; frac\; \{D\; \backslash \; vec\; \{AP\}\}\; \{dt\}$,
• the rotation of R' compared to R being represented by the vector $\backslash \; vec\; \{\backslash \; Omega\}$:

$\backslash \; frac\; \{D\; \backslash \; vec\; \{COp\}\}\; \{dt\}\; =\; \backslash \; frac\; \{D\; \backslash \; vec\; \{AP\}\}\; \{dt\}\; +\; (\backslash \; frac\; \{D\; \backslash \; vec\; \{OA\}\}\; \{dt\}\; +\; \backslash \; vec\; \{\backslash \; Omega\}\; \backslash \; wedge\; \backslash \; vec\; \{AP\})$, is: $\backslash \; vec\; \{v\}\; \_\; \{absolute\}\; =\; \backslash \; vec\; \{v\}\; \_\; \{relative\}\; +\; \backslash \; vec\; \{v\}\; \_\; \{drive\}$

It should be noticed that this mathematical operation does not imply any specificity for the reference mark R which one can describe as absolute compared to R', relative reference mark. It is not inevitably motionless.

This formula of change of reference mark, known as also of Composition of the movements making it possible to compare speeds and accelerations is however dependant on an important condition, that of a universal, valid stop watch in the various reference frames.

Change of reference frame in relativistic mechanics

August 1st

See too

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