All dissertations will be conducted only for one-dimensional model, i.e. all analyzed vector values will be parallel to *x*-axis. Each derived dynamic can easily be generalized into three-dimensional cases.

In order to derive dynamics in the Special Theory of Relativity (STR), it is necessary to adopt an additional assumption, which allows the concept of mass, momentum and kinetic energy to be introduced into the theory. Depending on the assumption, different dynamics of bodies are received.

The inertial mass body resting in inertial frame of reference is determined by *m*_{0} (rest mass). The rest mass is determined on the base unit of mass and the method of comparing any mass with this base unit. The inertial mass body at rest in *U*_{2}, as seen from *U*_{1} system, is determined by *m*_{2/1} (relativistic mass). It is worth to note that the relativistic mass in this case is an inertial mass that occurs in the Newton’s second law, rather than mass occurring in the formula for momentum, as assumed in the STR. In this way, a different definition of relativistic mass has been adopted, than one in the STR. Such a definition of the relativistic mass is more convenient in deriving dynamics.

The body of *m*_{0} inertial mass is in *U*_{2} system. It is affected by force *F*_{2/2} that causes acceleration of *dv*_{2/2}/*dt*_{2}. Therefore, for the observer from *U*_{2} system, the Newton’s second law takes a form of

$${F}_{2/2}:={m}_{0}\cdot {a}_{2/2}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}$$(24)For the observer from *U*_{1} system, inertial mass of the same body is *m*_{2/1}. For this observer, the force *F*_{2/1} acts on the body, causing acceleration of *dv*_{2/1}/*dt*_{1}. Therefore, for the observer from *U*_{1} the Newton’s second law takes the form of

$$\begin{array}{rl}{F}_{2/1}& :=f({v}_{2/1})\cdot {m}_{0}\cdot {a}_{2/1}={m}_{2/1}({v}_{2/1})\cdot {a}_{2/1}\\ & ={m}_{2/1}\cdot {a}_{2/1}={m}_{2/1}\frac{d{v}_{2/1}}{d{t}_{1}}\end{array}$$(25)Equation (25) means that a generalized form of the Newton’s second law is postulated. This generalized form contains an additional parameter *f*(*v*). From the formula (24) shows that always *f* (0) = 1. In classical mechanics *f*(*v*) = 1, while in the current dynamics STR *f*(*v*) = _{3} (formula (32)). Determining another form of parameter *f*(*v*) leads to other dynamics for STR. The inertial relativistic mass *m*_{2/1} is the product of this additional parameter *f*(*v*) and the inertial mass body at rest *m*_{0}. In this article, the parameter *f*(*v*) will not be used, only the inertial relativistic mass *m*_{2/1}.

Definitions identical as in classical mechanics apply for momentum and kinetic energy.

For the observer from *U*_{2} system, the change of this body momentum can be recorded in the following forms

$$\begin{array}{rl}d{p}_{2/2}& :={F}_{2/2}\cdot d{t}_{2}={m}_{0}\cdot {a}_{2/2}\cdot d{t}_{2}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}d{t}_{2}\\ & ={m}_{0}\cdot d{v}_{2/2}\end{array}$$(26)For the observer from *U*_{1} system, the change of this body momentum can be recorded in the following forms

$$\begin{array}{rl}d{p}_{2/1}& :={F}_{2/1}\cdot d{t}_{1}={m}_{2/1}\cdot {a}_{2/1}\cdot d{t}_{1}={m}_{2/1}\frac{d{v}_{2/1}}{d{t}_{1}}d{t}_{1}\\ & ={m}_{2/1}\cdot d{v}_{2/1}\end{array}$$(27)where:

–

*dp*_{2/2} is a change of body momentum with rest mass *m*_{0} in the inertial system *U*_{2}, measured by the observer from the same inertial system *U*_{2},

–

*dp*_{2/1} is a change of body momentum in the inertial system *U*_{2}, measured by the observer from the same inertial system *U*_{1}.

Kinetic energy of the body is equal of the work into its acceleration. For the observer from *U*_{1} system, the change of kinetic energy of this body is as follows

$$\begin{array}{rl}d{E}_{2/1}& :={F}_{2/1}\cdot d{x}_{2/1}={m}_{2/1}\cdot {a}_{2/1}\cdot d{x}_{2/1}\\ & ={m}_{2/1}\frac{d{v}_{2/1}}{d{t}_{1}}d{x}_{2/1}={m}_{2/1}\frac{d{x}_{2/1}}{d{t}_{1}}d{v}_{2/1}\\ & ={m}_{2/1}\cdot {v}_{2/1}\cdot d{v}_{2/1}\end{array}$$(28)where:

*– dE*_{2/1} is a change of kinetic energy of the body in inertial system *U*_{2}, measured by the observer from the inertial system *U*_{1}.

## 4.1 STR dynamics with constant force (STR/*F*)

In this section, a model of dynamics of bodies based on the assumption that the force accelerating of the body (parallel to *x*-axis) is the same for an observer from every inertial system will be derived (hence indication *F*).

## 4.1.1 The relativistic mass in STR/*F*

In the model STR/*F* it is assumed, that

$${F}_{2/1}^{F}:={F}_{2/2}$$(29)Having introduced (24) and (25), one obtains

$${m}_{2/1}^{F}\frac{d{v}_{2/1}}{d{t}_{1}}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}$$(30)On the base (20) and (23), one has

$${m}_{2/1}^{F}\frac{d{v}_{2/1}}{d{t}_{1}}={m}_{0}\frac{\frac{d{v}_{2/1}}{1-{({v}_{2/1}/c)}^{2}}}{\sqrt{1-{({v}_{2/1}/c)}^{2}}\cdot d{t}_{1}}$$(31)Hence, a formula for relativistic mass of the body that is located in the system *U*_{2} and is seen from the system *U*_{1} is obtained, when assumption (29) is satisfied, as below

$${m}_{2/1}^{F}={m}_{0}{\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}\right]}^{\phantom{\rule{1pt}{0ex}}3/2}$$(32)## 4.1.2 The momentum in STR/*F*

The body of rest mass *m*_{0} is associated with the system *U*_{2}. To determine the momentum of the body relative to the system *U*_{1} a substitution of (32) to (27)

$$\begin{array}{rl}d{p}_{2/1}^{F}& ={m}_{2/1}^{F}\cdot d{v}_{2/1}={m}_{0}{\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}\right]}^{3/2}d{v}_{2/1}\\ & ={m}_{0}{c}^{3}\frac{1}{{({c}^{2}-{v}_{2/1}^{2})}^{3/2}}d{v}_{2/1}\end{array}$$(33)The body momentum is a sum of increases in its momentum, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${p}_{2/1}^{F}={m}_{0}{c}^{3}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{1}{{({c}^{2}-{v}_{2/1}^{2})}^{3/2}}d{v}_{2/1}$$(34)From the work [2] (formula 72, p. *F*167) it is possible to read out, that

$$\int \frac{dx}{{({a}^{2}-{x}^{2})}^{3/2}}=\frac{x}{{a}^{2}\sqrt{{a}^{2}-{x}^{2}}},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}a\ne 0$$(35)After applying the integral (35) to (34) the formula for the body momentum in *U*_{2} system is received and measured by the observer from *U*_{1} system in a form of

$${p}_{2/1}^{F}={m}_{0}{c}^{3}\frac{{v}_{2/1}}{{c}^{2}\sqrt{{c}^{2}-{v}_{2/1}^{2}}}=\frac{{m}_{0}}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}{v}_{2/1}$$(36)This formula is identical to the formula for momentum known from the STR, for the same reasons as in the case of momentum. This is because the dynamics known from the STR is derived from the assumption (29). It was adopted unconsciously, because it was considered as necessary. The awareness of this assumption allows to its change and derives other dynamics.

As already mentioned above, the definition of relativistic mass adopted is different from the definition adopted in the STR. In this case, the relativistic mass is the one, which occurs in the Newton’s second law (25). In this particular case, it is expressed in terms of dependency (32). In the STR, the relativistic mass is the one, which occurs in the formula (36) per momentum.

## 4.1.3 The momentum in STR/*F* for small velocities

For small velocity *v*_{2/1} << *c* momentum (36) comes down to the momentum from classical mechanics, because

$${v}_{2/1}<<c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{p}_{2/1}^{F}\approx {m}_{0}{v}_{2/1}$$(37)## 4.1.4 The kinetic energy in STR/*F*

A determination of the formula for kinetic energy will be given. The dependence for the relativistic mass (32) is introduced to the formula (28)

$$\begin{array}{rl}d{E}_{2/1}^{F}& ={m}_{2/1}^{F}\cdot {v}_{2/1}\cdot d{v}_{2/1}={m}_{0}{\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}\right]}^{3/2}{v}_{2/1}d{v}_{2/1}\\ & ={m}_{0}{c}^{3}\frac{{v}_{2/1}}{{({c}^{2}-{v}_{2/1}^{2})}^{3/2}}d{v}_{2/1}\end{array}$$(38)The kinetic energy of body is a sum of increases in its kinetic energy, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${E}_{2/1}^{F}={m}_{0}{c}^{3}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{{v}_{2/1}}{{({c}^{2}-{v}_{2/1}^{2})}^{3/2}}d{v}_{2/1}$$(39)From the work [2] (formula 74, p. 167) it is possible to read out, that

$$\int \frac{xdx}{{({a}^{2}-{x}^{2})}^{3/2}}=\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}$$(40)After applying the integral (40) to (39) the formula for the kinetic energy of the body in *U*_{2} system and measured by the observer from *U*_{1} system in a form of

$$\begin{array}{rl}{E}_{2/1}^{F}& ={m}_{0}{c}^{3}{\left.\frac{1}{\sqrt{{c}^{2}-{x}^{2}}}\right|}_{0}^{{v}_{2/1}}={m}_{0}{c}^{3}\left(\frac{1}{\sqrt{{c}^{2}-{v}_{2/1}^{2}}}-\frac{1}{c}\right)\\ & ={m}_{0}{c}^{2}\frac{1}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}-{m}_{0}{c}^{2}\end{array}$$(41)This formula is identical to the formula for kinetic energy known from the STR, for the same reasons as in the case of momentum (36).

## 4.1.5 The kinetic energy in STR/*F* for small velocities

Formula (41) can be written in the form

$${E}_{2/1}^{F}={m}_{0}{c}^{2}\frac{1-\sqrt{1-{({v}_{2/1}/c)}^{2}}}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}\cdot \frac{1+\sqrt{1-{({v}_{2/1}/c)}^{2}}}{1+\sqrt{1-{({v}_{2/1}/c)}^{2}}}$$(42)$${E}_{2/1}^{F}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}\frac{2}{1-\frac{{v}_{2/1}^{2}}{{c}^{2}}+\sqrt{1-\frac{{v}_{2/1}^{2}}{{c}^{2}}}}\phantom{\rule{thickmathspace}{0ex}}$$(43)On this basis, for small values *v*_{2/1} *≪c* one receives

$${v}_{2/1}\ll c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{E}_{2/1}^{F}\approx \frac{{m}_{0}{v}_{2/1}^{2}}{2}\frac{2}{1+1}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}$$(44)## 4.1.6 The force in STR/*F*

Due to the assumption (29) value measurement of the same force by two different observers is identical.

## 4.2 STR dynamics with constant momentum change (STR/*Δp*)

In this section, a model of dynamics of bodies based on the assumption that the change in momentum of the body (parallel to *x*-axis) is the same for an observer from every inertial system will be derived (hence indication *Δp*).

These dynamics seem particularly interesting, because the conservation law of momentum is a fundamental law. Assumption that the change of body momentum is the same for every observer seems to be a natural extension of this law.

## 4.2.1 The relativistic mass in STR/*Δp*

In the model STR/*Δp* it is assumed, that

$$d{p}_{2/1}^{\mathrm{\Delta}p}:=d{p}_{2/2}$$(45)Having introduced (26) and (27), one obtains

$${m}_{2/1}^{\mathrm{\Delta}p}d{v}_{2/1}={m}_{0}d{v}_{2/2}$$(46)On the base (20), one has

$${m}_{2/1}^{\mathrm{\Delta}p}d{v}_{2/1}={m}_{0}\frac{d{v}_{2/1}}{1-{({v}_{2/1}/c)}^{2}}$$(47)Hence, a formula for relativistic mass of the body that is located in the system *U*_{2} and is seen from the system *U*_{1} is obtained, when assumption (45) is satisfied, as below

$${m}_{2/1}^{\mathrm{\Delta}p}={m}_{0}\frac{1}{1-{({v}_{2/1}/c)}^{2}}$$(48)## 4.2.2 The momentum in STR/*Δp*

The body of rest mass *m*_{0} is associated with the system *U*_{2}. To determine the momentum of the body relative to the system *U*_{1} a substitution of (48) to (27) is made

$$\begin{array}{rl}d{p}_{2/1}^{\mathrm{\Delta}p}& ={m}_{2/1}^{\mathrm{\Delta}p}\cdot d{v}_{2/1}={m}_{0}\frac{1}{1-{({v}_{2/1}/c)}^{2}}d{v}_{2/1}\\ & ={m}_{0}{c}^{2}\frac{1}{{c}^{2}-{v}_{2/1}^{2}}d{v}_{2/1}\end{array}$$(49)The body momentum is a sum of increases in its momentum, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${p}_{2/1}^{\mathrm{\Delta}p}={m}_{0}{c}^{2}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{1}{{c}^{2}-{v}_{2/1}^{2}}d{v}_{2/1}$$(50)From the work [2] (formula 52, p. 160) it is possible to read out, that

$$\int \frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\mathrm{ln}\left|\frac{a+x}{a-x}\right|,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}a\ne 0$$(51)After applying the integral (51) to (50) the formula for the body momentum in *U*_{2} system and measured by the observer from *U*_{1} system is received in a form of

$${p}_{2/1}^{\mathrm{\Delta}p}={m}_{0}{c}^{2}{\left.\frac{1}{2c}\mathrm{ln}\left|\frac{c+x}{c-x}\right|\right|}_{0}^{{v}_{2/1}}=\frac{{m}_{0}c}{2}\mathrm{ln}\left(\frac{c+{v}_{2/1}}{c-{v}_{2/1}}\right)$$(52)## 4.2.3 The momentum in STR/*Δp* for small velocities

Formula (52) can be written in the form

$$\begin{array}{rl}{p}_{2/1}^{\mathrm{\Delta}p}& =\frac{{m}_{0}{v}_{2/1}}{2}\frac{c}{{v}_{2/1}}\mathrm{ln}\left(\frac{c+{v}_{2/1}}{c-{v}_{2/1}}\right)\\ & =\frac{{m}_{0}{v}_{2/1}}{2}\mathrm{ln}\left(\frac{{(1+{v}_{2/1}/c)}^{c/{v}_{2/1}}}{{(1-{v}_{2/1}/c)}^{c/{v}_{2/1}}}\right)\end{array}$$(53)$${p}_{2/1}^{\mathrm{\Delta}p}=\frac{{m}_{0}{v}_{2/1}}{2}\mathrm{ln}\left(\frac{{\left(1+\frac{1}{c/{v}_{2/1}}\right)}^{c/{v}_{2/1}}}{{\left(1-\frac{1}{c/{v}_{2/1}}\right)}^{c/{v}_{2/1}}}\right)$$(54)On this basis, for small values *v*_{2/1} << *c* one receives

$$\begin{array}{rl}{v}_{2/1}\ll c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{p}_{2/1}^{\mathrm{\Delta}p}& \approx \frac{{m}_{0}{v}_{2/1}}{2}\mathrm{ln}\left(\frac{e}{1/e}\right)\\ & =\frac{{m}_{0}{v}_{2/1}}{2}\mathrm{ln}({e}^{2})={m}_{0}{v}_{2/1}\end{array}$$(55)## 4.2.4 The kinetic energy in STR/*Δp*

A determination of the formula for kinetic energy will be given. The dependence for the relativistic mass (48) is introduced to the formula (28)

$$\begin{array}{rl}d{E}_{2/1}^{\mathrm{\Delta}p}& ={m}_{2/1}^{\mathrm{\Delta}p}\cdot {v}_{2/1}\cdot d{v}_{2/1}={m}_{0}\frac{1}{1-{({v}_{2/1}/c)}^{2}}{v}_{2/1}d{v}_{2/1}\\ & ={m}_{0}{c}^{2}\frac{{v}_{2/1}}{{c}^{2}-{v}_{2/1}^{2}}d{v}_{2/1}\end{array}$$(56)The kinetic energy of body is a sum of increases in its kinetic energy, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${E}_{2/1}^{\mathrm{\Delta}p}={m}_{0}{c}^{2}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{{v}_{2/1}}{{c}^{2}-{v}_{2/1}^{2}}d{v}_{2/1}$$(57)From the work [2] (formula 56, p. 160) it is possible to read out, that

$$\int \frac{x}{{a}^{2}-{x}^{2}}dx=-\frac{1}{2}\mathrm{ln}\left|{a}^{2}-{x}^{2}\right|$$(58)After applying the integral (58) to (57) the formula for the kinetic energy of the body in *U*_{2} system and measured by the observer from *U*_{1} system in a form of

$$\begin{array}{rl}{E}_{2/1}^{\mathrm{\Delta}p}& =-{m}_{0}{c}^{2}{\left.\frac{1}{2}\mathrm{ln}\left|{c}^{2}-{x}^{2}\right|\right|}_{0}^{{v}_{2/1}}\\ & =-\frac{{m}_{0}{c}^{2}}{2}\mathrm{ln}({c}^{2}-{v}_{2/1}^{2})+\frac{{m}_{0}{c}^{2}}{2}\mathrm{ln}({c}^{2})\end{array}$$(59)$${E}_{2/1}^{\mathrm{\Delta}p}=\frac{{m}_{0}{c}^{2}}{2}\mathrm{ln}\frac{{c}^{2}}{{c}^{2}-{v}_{2/1}^{2}}=\frac{{m}_{0}{c}^{2}}{2}\mathrm{ln}\frac{1}{1-{({v}_{2/1}/c)}^{2}}$$(60)## 4.2.5 The kinetic energy in STR/*Δp* for small velocities

Formula (60) can be written in the form

$$\begin{array}{rl}{E}_{2/1}^{\mathrm{\Delta}p}& =\frac{{m}_{0}{v}_{2/1}^{2}}{2}\frac{{c}^{2}}{{v}_{2/1}^{2}}\mathrm{ln}\frac{1}{1-{({v}_{2/1}/c)}^{2}}\\ & =\frac{{m}_{0}{v}_{2/1}^{2}}{2}\mathrm{ln}\frac{1}{{[1-{({v}_{2/1}/c)}^{2}]}^{{(c/{v}_{2/1})}^{2}}}\end{array}$$(61)$${E}_{2/1}^{\mathrm{\Delta}p}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}\mathrm{ln}\frac{1}{{\left[1-\frac{1}{{(c/{v}_{2/1})}^{2}}\right]}^{{(c/{v}_{2/1})}^{2}}}$$(62)On this basis, for small values *v*_{2/1} << *c* one receives

$${v}_{2/1}\ll c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{E}_{2/1}^{\mathrm{\Delta}p}\approx \frac{{m}_{0}{v}_{2/1}^{2}}{2}\mathrm{ln}\frac{1}{1/e}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}$$(63)## 4.2.6 The force in STR/*Δp*

Body with rest mass *m*_{0} is related to *U*_{2} system. It is affected by force that causes acceleration. For the observer from this system, the acceleration force has in accordance with (24) the following value

$${F}_{2/2}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}$$(64)For the observer from *U*_{1} system, acceleration force has in accordance with (25) the following value

$${F}_{2/1}^{\mathrm{\Delta}p}={m}_{2/1}^{\mathrm{\Delta}p}\frac{d{v}_{2/1}}{d{t}_{1}}$$(65)If to divide parties’ equation (65) by (64), then on the basis of (20) and (23) one will receive

$$\frac{{F}_{2/1}^{\mathrm{\Delta}p}}{{F}_{2/2}}=\frac{{m}_{2/1}^{\mathrm{\Delta}p}}{{m}_{0}}\cdot \frac{d{t}_{2}}{d{t}_{1}}\cdot \frac{d{v}_{2/1}}{d{v}_{2/2}}=\frac{{m}_{2/1}^{\mathrm{\Delta}p}}{{m}_{0}}(1-({v}_{2/1}/c{)}^{2}{)}^{3/2}$$(66)On the basis of (48) a relation between measurements of the same force by two different observers is obtained

$${F}_{2/1}^{\mathrm{\Delta}p}=\sqrt{1-{({v}_{2/1}/c)}^{2}}\cdot {F}_{2/2}$$(67)The highest value of force is measured by the observer from the inertial system in which the body is located.

## 4.3 STR dynamics with constant mass (STR/*m*)

In this section, a model of dynamics of bodies, based on the assumption that body weight is the same for an observer from each inertial reference system, will be derived (hence indication *m*).

## 4.3.1 The relativistic mass in STR/*m*

In the model STR/*m* it is assumed, that

$${m}_{2/1}^{m}:={m}_{0}$$(68)Therefore, for the observer from inertial system *U*_{1}, the body mass in *U*_{2} system is the same as the rest mass.

## 4.3.2 The momentum in STR/*m*

The body of rest mass *m*_{0} is associated with the system *U*_{2}. To determine the momentum of the body relative to the system *U*_{1} a substitution of (68) to (27)

$$d{p}_{2/1}^{m}={m}_{2/1}^{m}\cdot d{v}_{2/1}={m}_{0}d{v}_{2/1}$$(69)The body momentum is a sum of increases in its momentum, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${p}_{2/1}^{m}={m}_{0}\underset{0}{\overset{{v}_{2/1}}{\int}}d{v}_{2/1}={m}_{0}{v}_{2/1}$$(70)In this relativistic dynamics the momentum is expressed with the same equation as in classical mechanics.

## 4.3.3 The kinetic energy in STR/*m*

A determination of the formula for kinetic energy will be given. The dependence for the relativistic mass (68) is introduced to the formula (28)

$$d{E}_{2/1}^{m}={m}_{2/1}^{m}\cdot {v}_{2/1}\cdot d{v}_{2/1}={m}_{0}{v}_{2/1}d{v}_{2/1}$$(71)The kinetic energy of body is a sum of increases in its kinetic energy, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${E}_{2/1}^{m}={m}_{0}\underset{0}{\overset{{v}_{2/1}}{\int}}{v}_{2/1}d{v}_{2/1}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}$$(72)In this relativistic dynamics the kinetic energy is expressed with the same equation as in classical mechanics.

## 4.3.4 The force in STR/*m*

Body with rest mass *m*_{0} is related to *U*_{2} system. It is affected by force that causes acceleration. For the observer from this system, the acceleration force has in accordance with (24) the following value

$${F}_{2/2}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}$$(73)For the observer from *U*_{1} system, acceleration force has in accordance with (25) the following value

$${F}_{2/1}^{m}={m}_{2/1}^{m}\frac{d{v}_{2/1}}{d{t}_{1}}={m}_{0}\frac{d{v}_{2/1}}{d{t}_{1}}$$(74)If to divide parties’ equation (74) by (73), then on the basis of (20) and (23) one will receive

$$\frac{{F}_{2/1}^{m}}{{F}_{2/2}}=\frac{d{t}_{2}}{d{t}_{1}}\cdot \frac{d{v}_{2/1}}{d{v}_{2/2}}=(1-({v}_{2/1}/c{)}^{2}{)}^{3/2}$$(75)i.e.

$${F}_{2/1}^{m}=(1-({v}_{2/1}/c{)}^{2}{)}^{3/2}\cdot {F}_{2/2}$$(76)The highest value of force is measured by the observer from the inertial system in which the body is located.

## 4.3.5 Discussion on the STR/*m* dynamics

Obtaining a relativistic dynamics, in which there is no relativistic mass, and equations for kinetic energy and momentum are identical as in classical mechanics can be surprising, because in relativistic mechanics it is believed that the accelerated body can achieve maximum speed *c*. However, this dynamics is formally correct.

If the body velocity *v*_{2/1} reaches *c* value, then according to (76)

$${F}_{2/1}^{m}=(1-{1}^{-}{)}^{3/2}\cdot {F}_{2/2}\approx 0$$(77)In the inertial system *U*_{2}, in which the body is located, can be affected by acceleration force *F*_{2/2} of any, but finite value. However, from a perspective of the inertial system *U*_{1}, towards which the body has *c* velocity, the same force is zero. This means that from a perspective of *U*_{1} system, it is not possible to perform work on the body, which will increase its kinetic energy indefinitely. From the relation (72) it results that the kinetic energy, that a body with mass *m*_{0} and velocity *c* has, a value has

$${E}_{max}^{m}=\frac{{m}_{0}{c}^{2}}{2}$$(78)## 4.4 STR dynamics with constant force to its operation time (STR/*F* /*Δt*)

In this section, a model of dynamics of bodies based on the assumption that the force that accelerates of the body (parallel to *x*-axis) divided by the time of operation of this force is the same for an observer from every inertial system will be derived (hence indication *F*/*Δt*).

## 4.4.1 The relativistic mass in STR/*F*/*Δt*

In the model STR/*F*/*Δt* it is assumed, that

$$\frac{{F}_{2/1}^{F/\mathrm{\Delta}t}}{d{t}_{1}}:=\frac{{F}_{2/2}}{d{t}_{2}}$$(79)Having introduced (24) and (25), one obtains

$${m}_{2/1}^{F/\mathrm{\Delta}t}\frac{d{v}_{2/1}}{d{t}_{1}}\frac{1}{d{t}_{1}}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}\frac{1}{d{t}_{2}}$$(80)On the base (20) and (23), one has

$${m}_{2/1}^{F/\mathrm{\Delta}t}\frac{d{v}_{2/1}}{d{t}_{1}^{2}}={m}_{0}\frac{\frac{d{v}_{2/1}}{1-{({v}_{2/1}/c)}^{2}}}{(1-{({v}_{2/1}/c)}^{2})d{t}_{1}^{2}}$$(81)Hence, a formula for relativistic mass of the body that is located in the system *U*_{2} and is seen from the system *U*_{1} is obtained, when assumption (79) is satisfied, as below

$${m}_{2/1}^{F/\mathrm{\Delta}t}={m}_{0}{\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}\right]}^{\phantom{\rule{1pt}{0ex}}2}$$(82)## 4.4.2 The momentum in STR/*F*/*Δt*

The body of rest mass *m*_{0} is associated with the system *U*_{2}. To determine the momentum of the body relative to the system *U*_{1} a substitution of (82) to (27)

$$\begin{array}{rl}d{p}_{2/1}^{F/\mathrm{\Delta}t}& ={m}_{2/1}^{F/\mathrm{\Delta}t}\cdot d{v}_{2/1}={m}_{0}{\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}\right]}^{\phantom{\rule{1pt}{0ex}}2}d{v}_{2/1}\\ & ={m}_{0}{c}^{4}\frac{1}{{({c}^{2}-{v}_{2/1}^{2})}^{2}}d{v}_{2/1}\end{array}$$(83)The body momentum is a sum of increases in its momentum, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${p}_{2/1}^{F/\mathrm{\Delta}t}={m}_{0}{c}^{4}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{1}{{({c}^{2}-{v}_{2/1}^{2})}^{2}}d{v}_{2/1}$$(84)From the work [2] (formula 54, p. 160) it is possible to read out, that

$$\int \frac{dx}{{({a}^{2}-{x}^{2})}^{2}}=\frac{x}{2{a}^{2}({a}^{2}-{x}^{2})}+\frac{1}{4{a}^{3}}\mathrm{ln}\left|\frac{a+x}{a-x}\right|,\phantom{\rule{1em}{0ex}}a\ne 0$$(85)After applying the integral (85) to (84) the formula for the body momentum in *U*_{2} system and measured by the observer from *U*_{1} system in a form of

$$\begin{array}{rl}{p}_{2/1}^{F/\mathrm{\Delta}t}& ={m}_{0}{c}^{4}{\left.\left[\frac{x}{2{c}^{2}({c}^{2}-{x}^{2})}+\frac{1}{4{c}^{3}}\mathrm{ln}\frac{(c+x)}{(c-x)}\right]\right|}_{0}^{{v}_{2/1}}\\ & ={m}_{0}c\left[\frac{c{v}_{2/1}}{2({c}^{2}-{v}_{2/1}^{2})}+\frac{1}{4}\mathrm{ln}\frac{(c+{v}_{2/1})}{(c-{v}_{2/1})}\right]\end{array}$$(86)$${p}_{2/1}^{F/\mathrm{\Delta}t}={m}_{0}{v}_{2/1}\frac{1}{2}\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}+\mathrm{ln}{\left(\frac{c+{v}_{2/1}}{c-{v}_{2/1}}\right)}^{\frac{c}{2{v}_{2/1}}}\right]$$(87)## 4.4.3 The momentum in STR/*F*/*Δt* for small velocities

Formula (87) can be written in the form

$$\begin{array}{rl}& {p}_{2/1}^{F/\mathrm{\Delta}t}=\\ & {m}_{0}{v}_{2/1}\left[\frac{1}{2(1-{({v}_{2/1}/c)}^{2})}+\frac{1}{4}\mathrm{ln}\left(\frac{{(1+{v}_{2/1}/c)}^{c/{v}_{2/1}}}{{(1-{v}_{2/1}/c)}^{c/{v}_{2/1}}}\right)\right]\end{array}$$(88)$$\begin{array}{rl}& {p}_{2/1}^{F/\mathrm{\Delta}t}=\\ & {m}_{0}{v}_{2/1}\left[\frac{1}{2(1-{({v}_{2/1}/c)}^{2})}+\frac{1}{4}\mathrm{ln}\left(\frac{{\left(1+\frac{1}{c/{v}_{2/1}}\right)}^{c/{v}_{2/1}}}{{\left(1-\frac{1}{c/{v}_{2/1}}\right)}^{c/{v}_{2/1}}}\right)\right]\end{array}$$(89)On this basis, for small values *v*_{2/1} << *c* one receives

$$\begin{array}{rl}{v}_{2/1}\ll c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{p}_{2/1}^{F/\mathrm{\Delta}t}& \approx {m}_{0}{v}_{2/1}\left[\frac{1}{2}+\frac{1}{4}\mathrm{ln}\left(\frac{e}{1/e}\right)\right]\\ & ={m}_{0}{v}_{2/1}\left[\frac{1}{2}+\frac{1}{4}\mathrm{ln}({e}^{2})\right]={m}_{0}{v}_{2/1}\end{array}$$(90)## 4.4.4 The kinetic energy in STR/*F*/*Δt*

A determination of the formula for kinetic energy will be given. The dependence for the relativistic mass (82) is introduced to the formula (28)

$$\begin{array}{rl}d{E}_{2/1}^{F/\mathrm{\Delta}t}& ={m}_{2/1}^{F/\mathrm{\Delta}t}\cdot {v}_{2/1}\cdot d{v}_{2/1}\\ & ={m}_{0}{\left[\frac{1}{1-{({v}_{2/1}/c)}^{2}}\right]}^{\phantom{\rule{1pt}{0ex}}2}{v}_{2/1}d{v}_{2/1}\\ & ={m}_{0}{c}^{4}\frac{{v}_{2/1}}{{({c}^{2}-{v}_{2/1}^{2})}^{2}}d{v}_{2/1}\end{array}$$(91)The kinetic energy of body is a sum of increases in its kinetic energy, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${E}_{2/1}^{F/\mathrm{\Delta}t}={m}_{0}{c}^{4}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{{v}_{2/1}}{{({c}^{2}-{v}_{2/1}^{2})}^{2}}d{v}_{2/1}$$(92)From the work [2] (formula 58, p. 160) it is possible to read out, that

$$\int \frac{xdx}{{({a}^{2}-{x}^{2})}^{2}}=\frac{1}{2({a}^{2}-{x}^{2})}$$(93)After applying the integral (93) do (92) the formula for the kinetic energy of the body in *U*_{2} system and measured by the observer from *U*_{1} system in a form of

$${E}_{2/1}^{F/\mathrm{\Delta}t}={m}_{0}{c}^{4}{\left.\frac{1}{2({c}^{2}-{x}^{2})}\right|}_{0}^{{v}_{2/1}}=\frac{{m}_{0}{c}^{4}}{2}\frac{1}{({c}^{2}-{v}_{2/1}^{2})}-\frac{{m}_{0}{c}^{4}}{2}\frac{1}{{c}^{2}}$$(94)$${E}_{2/1}^{F/\mathrm{\Delta}t}=\frac{{m}_{0}{c}^{2}}{2}\frac{1}{1-{({v}_{2/1}/c)}^{2}}-\frac{{m}_{0}{c}^{2}}{2}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}\frac{1}{1-{({v}_{2/1}/c)}^{2}}$$(95)The formula for kinetic energy (95) was derived from the work [3], due to the fact that the author adopted a different assumption than the one on which the dynamics known from the STR was based.

## 4.4.5 The kinetic energy in STR/*F*/*Δt* for small velocities

For small velocity *v*_{2/1}*≪c* kinetic energy (95) comes down to the kinetic energy from classical mechanics, because

$${v}_{2/1}\ll c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{E}_{2/1}^{F/\mathrm{\Delta}t}\approx \frac{{m}_{0}{v}_{2/1}^{2}}{2}\cdot \frac{1}{1}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}$$(96)## 4.4.6 The force in STR/*F*/*Δt*

Body with rest mass *m*_{0} is related to *U*_{2} system. It is affected by force that causes acceleration. For the observer from this system, the acceleration force has in accordance with (24) the following value

$${F}_{2/2}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}$$(97)For the observer from *U*_{1} system, acceleration force has in accordance with (25) the following value

$${F}_{2/1}^{F/\mathrm{\Delta}t}={m}_{2/1}^{F/\mathrm{\Delta}t}\frac{d{v}_{2/1}}{d{t}_{1}}$$(98)If to divide parties’ equation (98) by (97), then on the basis of (20) and (23) one will receive

$$\frac{{F}_{2/1}^{F/\mathrm{\Delta}t}}{{F}_{2/2}}=\frac{{m}_{2/1}^{F/\mathrm{\Delta}t}}{{m}_{0}}\cdot \frac{d{t}_{2}}{d{t}_{1}}\cdot \frac{d{v}_{2/1}}{d{v}_{2/2}}=\frac{{m}_{2/1}^{F/\mathrm{\Delta}t}}{{m}_{0}}(1-({v}_{2/1}/c{)}^{2}{)}^{3/2}$$(99)On the basis of (82) relation between measurements of the same force by two different observers is obtained

$${F}_{2/1}^{F/\mathrm{\Delta}t}=\frac{1}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}\cdot {F}_{2/2}$$(100)The lowest value of force is measured by the observer from the inertial system in which the body is located.

## 4.5 STR dynamics with constant mass to elapse of observer’s time (STR/*m*/*Δt*)

In this subchapter a model of body dynamics will be derived based on the assumption that the body mass divided by the elapse of time in observer system is the same for the observer from each inertial frame of reference (hence indication *m*/*Δt*).

## 4.5.1 The relativistic mass in STR/*m*/*Δt*

In the model STR/*m*/*Δt* it is assumed, that

$$\frac{{m}_{2/1}^{m/\mathrm{\Delta}t}}{d{t}_{1}}:=\frac{{m}_{0}}{d{t}_{2}}$$(101)On the base (23), one obtains

$$\frac{{m}_{2/1}^{m/\mathrm{\Delta}t}}{d{t}_{1}}=\frac{{m}_{0}}{\sqrt{1-{({v}_{2/1}/c)}^{2}}\cdot d{t}_{1}}$$(102)Hence, a formula for relativistic mass of the body that is located in the system *U*_{2} and is seen from the system *U*_{1} is obtained, when assumption (101) is satisfied, as below

$${m}_{2/1}^{m/\mathrm{\Delta}t}={m}_{0}\frac{1}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}$$(103)## 4.5.2 The momentum in STR/*m*/*Δt*

The body of rest mass *m*_{0} is associated with the system *U*_{2}. To determine the momentum of the body relative to the system *U*_{1} a substitution of (103) to (27)

$$\begin{array}{rl}d{p}_{2/1}^{m/\mathrm{\Delta}t}& ={m}_{2/1}^{m/\mathrm{\Delta}t}\cdot d{v}_{2/1}={m}_{0}\frac{1}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}d{v}_{2/1}\\ & ={m}_{0}c\frac{1}{\sqrt{{c}^{2}-{v}_{2/1}^{2}}}d{v}_{2/1}\end{array}$$(104)The body momentum is a sum of increases in its momentum, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${p}_{2/1}^{m/\mathrm{\Delta}t}={m}_{0}{c}^{2}\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{1}{\sqrt{{c}^{2}-{v}_{2/1}^{2}}}d{v}_{2/1}$$(105)From the work [2] (formula 71, p. 167) it is possible to read out, that

$$\int \frac{dx}{\sqrt{{a}^{2}-{x}^{2}}}=\mathrm{arcsin}\frac{x}{a},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}a>0$$(106)After applying the integral (106) to (105) the formula for the body momentum in *U*_{2} system and measured by the observer from *U*_{1} system in a form of

$${p}_{2/1}^{m/\mathrm{\Delta}t}={m}_{0}c\cdot {\left.\mathrm{arcsin}\frac{{v}_{2/1}}{c}\right|}_{0}^{{v}_{2/1}}={m}_{0}c\cdot \mathrm{arcsin}\frac{{v}_{2/1}}{c}$$(107)## 4.5.3 The momentum in STR/*m*/*Δt* for small velocities

Formula (107) can be written in the form

$${p}_{2/1}^{m/\mathrm{\Delta}t}={m}_{0}{v}_{2/1}\frac{\mathrm{arcsin}\frac{{v}_{2/1}}{c}}{\frac{{v}_{2/1}}{c}}$$(108)On this basis, for small values *v*_{2/1} << *c* one receives

$${v}_{2/1}\ll c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{p}_{2/1}^{m/\mathrm{\Delta}t}\approx {m}_{0}{v}_{2/1}$$(109)## 4.5.4 The kinetic energy in STR/*m*/*Δt*

A determination of the formula for kinetic energy will be given. The dependence for the relativistic mass (103) is introduced to the formula (28)

$$\begin{array}{rl}d{E}_{2/1}^{m/\mathrm{\Delta}t}& ={m}_{2/1}^{m/\mathrm{\Delta}t}\cdot {v}_{2/1}\cdot d{v}_{2/1}={m}_{0}\frac{1}{\sqrt{1-{({v}_{2/1}/c)}^{2}}}{v}_{2/1}d{v}_{2/1}\\ & ={m}_{0}c\frac{{v}_{2/1}}{\sqrt{{c}^{2}-{v}_{2/1}^{2}}}d{v}_{2/1}\end{array}$$(110)The kinetic energy of body is a sum of increases in its kinetic energy, when the body is accelerated from the inertial system *U*_{1} (the body has velocity 0) to the inertial system *U*_{2} (the body has velocity *v*_{2/1}), i.e.

$${E}_{2/1}^{m/\mathrm{\Delta}t}={m}_{0}c\underset{0}{\overset{{v}_{2/1}}{\int}}\frac{{v}_{2/1}}{\sqrt{{c}^{2}-{v}_{2/1}^{2}}}d{v}_{2/1}$$(111)From the work [2] (formula 73, p. 167) it is possible to read out, that

$$\int \frac{x}{\sqrt{{a}^{2}-{x}^{2}}}dx=-\sqrt{{a}^{2}-{x}^{2}}$$(112)After applying the integral (112) do (111) the formula for the kinetic energy of the body in *U*_{2} system and measured by the observer from *U*_{1} system in a form of

$${E}_{2/1}^{m/\mathrm{\Delta}t}=-{m}_{0}c{\left.\sqrt{{c}^{2}-{v}_{2/1}^{2}}\right|}_{0}^{{v}_{2/1}}=-{m}_{0}c\sqrt{{c}^{2}-{v}_{2/1}^{2}}+{m}_{0}c\sqrt{{c}^{2}}$$(113)$${E}_{2/1}^{m/\mathrm{\Delta}t}={m}_{0}{c}^{2}-{m}_{0}c\sqrt{{c}^{2}-{v}_{2/1}^{2}}={m}_{0}{c}^{2}(\phantom{\rule{thinmathspace}{0ex}}1-\sqrt{1-{({v}_{2/1}/c)}^{2}}\phantom{\rule{thinmathspace}{0ex}})$$(114)## 4.5.5 The kinetic energy in STR/*m*/*Δt* for small velocities

Formula (114) can be written in the form

$$\begin{array}{rl}& {E}_{2/1}^{m/\mathrm{\Delta}t}=\\ & \frac{{m}_{0}{v}_{2/1}^{2}}{2}\cdot \frac{2{c}^{2}}{{v}_{2/1}^{2}}\cdot \frac{(1-\sqrt{1-{({v}_{2/1}/c)}^{2}})(1+\sqrt{1-{({v}_{2/1}/c)}^{2}})}{1+\sqrt{1-{({v}_{2/1}/c)}^{2}}}\end{array}$$(115)$$\begin{array}{rl}{E}_{2/1}^{m/\mathrm{\Delta}t}& =\frac{{m}_{0}{v}_{2/1}^{2}}{2}\cdot \frac{2{c}^{2}}{{v}_{2/1}^{2}}\cdot \frac{1-(1-{({v}_{2/1}/c)}^{2})}{1+\sqrt{1-{({v}_{2/1}/c)}^{2}}}\\ & =\frac{{m}_{0}{v}_{2/1}^{2}}{2}\frac{2}{1+\sqrt{1-{({v}_{2/1}/c)}^{2}}}\end{array}$$(116)On this basis, for small values *v*_{2/1} *≪c* one receives

$${v}_{2/1}<<c\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{E}_{2/1}^{m/\mathrm{\Delta}t}\approx \frac{{m}_{0}{v}_{2/1}^{2}}{2}\cdot \frac{2}{2}=\frac{{m}_{0}{v}_{2/1}^{2}}{2}$$(117)## 4.5.6 The force in STR/*m*/*Δt*

Body with rest mass *m*_{0} is related to *U*_{2} system. It is affected by force that causes acceleration. For the observer from this system, the acceleration force has in accordance with (24) the following value

$${F}_{2/2}={m}_{0}\frac{d{v}_{2/2}}{d{t}_{2}}$$(118)For the observer from *U*_{1} system, acceleration force has in accordance with (25) the following value

$${F}_{2/1}^{m/\mathrm{\Delta}t}={m}_{2/1}^{m/\mathrm{\Delta}t}\frac{d{v}_{2/1}}{d{t}_{1}}$$(119)If to divide parties’ equation (119) by (118), then on the basis of (20) and (23) one will receive

$$\frac{{F}_{2/1}^{m/\mathrm{\Delta}t}}{{F}_{2/2}}=\frac{{m}_{2/1}^{m/\mathrm{\Delta}t}}{{m}_{0}}\cdot \frac{d{t}_{2}}{d{t}_{1}}\cdot \frac{d{v}_{2/1}}{d{v}_{2/2}}=\frac{{m}_{2/1}^{m/\mathrm{\Delta}t}}{{m}_{0}}(1-({v}_{2/1}/c{)}^{2}{)}^{3/2}$$(120)On the basis of (103) relation between measurements of the same force by two different observers is obtained

$${F}_{2/1}^{m/\mathrm{\Delta}t}=(1-({v}_{2/1}/c{)}^{2})\cdot {F}_{2/2}$$(121)The highest value of force is measured by the observer from the inertial system in which the body is located.

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