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# Skládání rychlostí v terorii relativity

<Frame>
  <img src="https://mintcdn.com/hogofogo0-cheesecakecorp-org-1857426d/iLkNrUdL5DB9CHdF/images/skladani1.png?fit=max&auto=format&n=iLkNrUdL5DB9CHdF&q=85&s=031a7f0a265f1bc0690c8dd699f4c8df" alt="Skladani1" width="2360" height="1640" data-path="images/skladani1.png" />
</Frame>

* $v = 20 \, \text{km} \cdot \text{h}^{-1}$ - rychlost vagónu
* $u' = 5 \, \text{km} \cdot \text{h}^{-1}$ - rychlost uživatele vagónu ve vagónu

**Klasická fyzika:** $u = v + u' = 20 + 5 = 25 \, \text{km} \cdot \text{h}^{-1}$

<Accordion title="Příklady, proč to někdy nefunguje">
  * $v = \frac{2}{3}c, u' = \frac{2}{3}c, u = \frac{4}{3}c$
  * $v = c, u' = c, u = 2c$

  → too bad
</Accordion>

**Spešl fyzika:**

<Frame>
  <img src="https://mintcdn.com/hogofogo0-cheesecakecorp-org-1857426d/iLkNrUdL5DB9CHdF/images/skladani2.png?fit=max&auto=format&n=iLkNrUdL5DB9CHdF&q=85&s=847581805f04dcf286f091778c2d341c" alt="Skladani2" width="2360" height="1640" data-path="images/skladani2.png" />
</Frame>

* v $S'$:
  * od $t_1'$ do $t_2'$ a od $x_1'$ do $x_2'$
  * $u' = \frac{x_2' - x_1'}{t_2' - t_1'}$
* v $S$:
  * $t_1$... $t_2$, $x_1$... $x_2$
  * $u = \frac{x_2 - x_1}{t_2 - t_1}$
  * $u = \frac{x_2 - x_1}{t_2 - t_1} = \frac{(x_2' + vt_2') - (x_1' + vt_1')}{(t_2' + \frac{v}{c^2}x_2')-(t_1' + \frac{v}{c^2}x_1')} = \frac{(x_2' - x_1')+v(t_2'-t_1')}{(t_2'-t_1')+\frac{v}{c^2}(x_2'-x_1')} = \frac{u' + v}{1 + \frac{u'v}{c^2}}$

$$
\boldsymbol{u = \frac{u' + v}{1 + \frac{v}{c^2}u'}}
$$

$v \rightarrow 0$: jmenovatel bude blízko $1$, a bude platit $u = u' + v$

<Accordion title="Oprava těch, co nefungují">
  * $v = \frac{2}{3}c, u' = \frac{2}{3}c, u = \frac{\frac{4}{3}c}{1 + \frac{\frac{4}{9}c^2}{c^2}} = \frac{12}{13}c$
  * $v = c, u' = c, u = \frac{2c}{1 + \frac{c^2}{c^2}} = c$
</Accordion>


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