# Speed

## Definition

• In physics, the speed is a size which measures the report/ratio of an evolution at time. Example: speed of Sedimentation, speed of a Chemical reaction, etc

• In kinematics, speed is a vector quantity which measures for a movement, the report/ratio of the distance covered at time.

## Speed in kinematics

One distinguishes:

• the curvilinear speed , which is the distance $d$ traversed on a Courbe per unit of Temps $t$. It is a size Scalaire.
$v = \ frac \left\{D\right\} \left\{T\right\}$
• the vector-speed or the speed in the space , which is the Vecteur

$\ vec \left\{v\right\} = \ frac \left\{\ vec \left\{\ mathrm Dr.\right\}\right\} \left\{\ mathrm dt\right\}$
whose standard is worth the speed and whose directions and direction are those of the movement of the object considered. Formally, vector-speed is the Dérivée from the position of the object compared to the Temps. When that does not involve confusions, one calls simply vector-speed “speed”. It is a vector quantity here.

The international unit speed is the Mètre by second (m.s-1). For the motor vehicles, one also frequently uses the kilometer hour (km/h), the Anglo-Saxon system uses the thousand per hour ( mile per hour , mph). In the navy, one uses the node, which is worth one thousand sailor per hour, that is to say 0,514  4  m.s-1. In aviation, one uses sometimes the Mach, Mach 1 being the speed of the his (which varies according to the temperature and from the pressure).

## History of the concept speed

A formal definition missed a long time with the concept speed, because the mathematicians avoided making the quotient of two nonhomogeneous sizes. To divide a distance by a time thus appeared as false to them as could seem to us today the sum of these two values. Thus to know if a body went more quickly than another, Galileo (1564-1642) compared the ratio of the distances covered by these bodies with the report/ratio of corresponding time. It applied for that following equivalence:

$\ frac \left\{s_1\right\} \left\{s_2\right\} \ the \ frac \left\{t_1\right\} \left\{t_2\right\} \ Leftrightarrow \ frac \left\{s_1\right\} \left\{t_1\right\} \ the \ frac \left\{s_2\right\} \left\{t_2\right\}$

The concept instantaneous speed is formally defined for the first time by Pierre Varignon (1654-1722) the July 5th 1698, like the report/ratio an infinitely small length $\ mathrm dx$ over time infinitely small $\ mathrm dt$ put to traverse this length. It earlier uses for that the formalism of the differential Calculus developped at the item fourteen years by Leibniz (1646-1716).

## The concept speed

It is necessary to distinguish two types of speed:

• the mean velocity, which answers the elementary definition very precisely. It is calculated by dividing the distance covered by run time; it has a direction over a given period;
• the instantaneous speed, which is obtained by passage to the limit of the definition speed. It is defined in one precise moment, via the concept of derivation $v = \ tfrac \left\{\ partial R\right\} \left\{\ partial T\right\}$. For example in calculations of Kinematic, speed is a Vecteur obtained by deriving the Cartesian Coordonnées from the position compared to time:
$\ vec \left\{v\right\} = \ frac \left\{\ partial \ vec \left\{R\right\}\right\} \left\{\ partial T\right\} = \ begin \left\{pmatrix\right\} \ frac \left\{\ partial X\right\} \left\{\ partial T\right\} \ \ \ frac \left\{\ partial there\right\} \left\{\ partial T\right\} \ \ \ frac \left\{\ partial Z\right\} \left\{\ partial T\right\} \ end \left\{pmatrix\right\}$

## Vector-speed

Instantaneous vector-speed $\ vec v$ of an object whose position at time $t$ is given by $\ vec X \left(T\right)$ calculated like the Dérivée

$\ vec v = \ frac \left\{\ mathrm D \ vec X\right\} \left\{\ mathrm dt\right\}$

Acceleration is the derivative speed, and speed is the derivative of the distance according to time.

The Accélération is the rate of shifting of speed of an object over the period. The average acceleration $a$ of an object of which speed changes starting from $v_i$ with $v_f$ for one period $t$ is given by:

$has = \ frac \left\{v_f - v_i\right\} t$

The vector of instantaneous acceleration $\ vec a$ of an object whose position at time $t$ is given by $\ vec X \left(T\right)$ is

$\ vec has = \ frac \left\{\ mathrm D \ vec v\right\} \left\{\ mathrm dt\right\} = \ frac \left\{\ mathrm d^2 \ vec X\right\} \left\{\ mathrm dt^2\right\}$

The final speed $v_f$ of an object starting with speed $v_i$ then accelerating with a constant rate $a$ during a time $t$ is:

$v_f = v_i + has T \,$

The mean velocity of an object undergoing a constant acceleration is $\left\{\ scriptstyle \ frac12\right\} \left(v_i + v_f\right)$. To find $d$ displacement of such an object accelerating for the period $t$, to substitute this expression in the first formula to obtain:

$D = T \ times \ frac \left\{v_i + v_f\right\} 2$

When only the initial swiftness of the object is known, the expression

$D = v_i T + \ frac \left\{has t^2\right\} 2$

can be used. These basic equations for the final swiftness and displacement can be combined to form an equation which is independent of time:

$v_f^2 = v_i^2 + 2 has d$

The equations above are valid for at the same time the traditional Mécanique but not for the restricted Relativité. In particular in traditional mechanics, all will be of agreement on the value of $t$ and the rules of transformation for the position create a situation in which all the observers not accelerating would describe the acceleration of an object with the same values. Neither one nor the other are true for restricted relativity.

The kinetic energy of a moving object is linear with its Masse and the square its speed:

$E_c = \ tfrac1 2 mv^2$

The kinetic energy is a quantity Scalaire.

### Polar coordinates

In Coordinated polar, speed in the plan can be broken up of radial speed, $\ mathrm Dr. \ mathrm dt$, moving away or going towards the origin and speed orthoradiale, in the perpendicular direction (which one will not confuse with the tangential component), equal to $r \ tfrac \left\{\ mathrm D \ theta\right\} \left\{\ mathrm dt\right\}$ (see angular Velocity).

The Angular momentum in the plan is

$\ vec L= m \ \ vec R \ wedge \ vec V = m \; r^2 \; \ frac \left\{\ mathrm D \ theta\right\} \left\{\ mathrm D T\right\} \ vec k$.

One recognizes in

$\ frac \left\{1\right\} \left\{2\right\} r^2 \ frac \left\{\ mathrm D \ theta\right\} \left\{\ mathrm D T\right\} = \ frac \left\{\ mathrm D has \left(T\right)\right\}\left\{\ mathrm D T\right\}$
areal speed.

If the force is central (see Mouvement with central force), then areal speed is constant (second law of Kepler).

## External bond

• Site of conversion of units (of which speed)

Simple: Speed Simple: Velocity

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