From 9e91c3bd6f63a7bbdfe8c6af39d10e17b5ab84bc Mon Sep 17 00:00:00 2001 From: Filip Znachor Date: Tue, 16 May 2023 09:51:55 +0200 Subject: [PATCH] =?UTF-8?q?P=C5=99id=C3=A1n=C3=AD=201.=20p=C5=99=C3=ADklad?= =?UTF-8?q?u=20z=20FYI?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- KFY FYI1/Příklad 1.md | 54 ++++++++++++++++++++++++++++++++++ KFY FYI1/_assets/priklad1.jpg | Bin 0 -> 17510 bytes 2 files changed, 54 insertions(+) create mode 100644 KFY FYI1/Příklad 1.md create mode 100644 KFY FYI1/_assets/priklad1.jpg diff --git a/KFY FYI1/Příklad 1.md b/KFY FYI1/Příklad 1.md new file mode 100644 index 0000000..f3a6651 --- /dev/null +++ b/KFY FYI1/Příklad 1.md @@ -0,0 +1,54 @@ +### Zadání + +Vlak se pohybuje po kruhové dráze o poloměru **800 m**. V počátečním okamžiku měl vlak rychlost **54 km/h** a v koncovém **18 km/h**. Mezi počátečním a koncovým okamžikem vlak urazil **800 m**. Určete: dobu potřebnou k uražení této dráhy a velikost zrychlení v počátečním a koncovém okamžiku. + +- $R = 800 \text{ m}$ +- $v_{0} = 54 \text{ km/h}$ +- $v_{1} = 18 \text{ km/h}$ +- $S = 800 \text{ m}$ +- $T = ?$ +- $a_{0} = ?$ +- $a_{1} = ?$ + +![](_assets/priklad1.jpg) + + +- $a = \sqrt{ a_{t}^2 + a_{n}^2 }$ +- $a_{t} = \text{konst.}$ (křivočarý pohyb rovnoměrně zrychlený) +- $\displaystyle a_{n} = \frac{v^2}{R} = \frac{v_{0}^2}{R}, \frac{v_{1}^2}{R}$ + +### Výpočet + +- $v = a_{t} \cdot t + v_{0}$ +- $s = \frac{1}{2}a_{t} \cdot t^2 + v_{0} \cdot t + s_{0}$ ++ pro $t = 0 \implies s_{0} = 0$ ++ $v = a_{t} \cdot t + v_{0}$ ++ $s = \frac{1}{2}a_{t} \cdot t^2 + v_{0} \cdot t$ +- pro $t = T$ +- $v_{1} = a_{t} \cdot T + v_{0}$ +- $s = \frac{1}{2}a_{t} \cdot T^2 + v_{0} \cdot T$ ++ $T = \frac{v_{1}-v_{0}}{a_{t}}$ + +--- + +$\displaystyle s = \frac{1}{2}\cancel{a_{t}} \cdot \frac{(v_{1} - v_{2})^T}{a_{t}^{\cancel{2}}} + v_{0} \cdot \frac{v_{1} - v_{0}}{a_{t}} = \frac{v_{1}^2 - 2v_{1}v_{0} + v_{0}^2}{2a_{t}} + \frac{v_{0}v_{1} - v_{0}^2}{a_{t}} = \frac{v_{1}^2 - \cancel{2v_{1}v_{0}} + \cancel{v_{0}^2} + \cancel{2v_{1}v_{0}} - \cancel{2}v_{0}^2}{2a_{t}} = \frac{v_{1}^2 - v_{0}^2}{2a_{t}}$ + +$\displaystyle T = \frac{v_{1} - v_{0}}{\frac{v_{1}^2 - v_{0}^2}{2s}} = \frac{v_{1} - v_{0}}{v_{1}^2 - v_{0}^2} \cdot 2s = \frac{\cancel{v_{1} - v_{0}}}{\cancel{(v_{1} - v_{0})}(v_{1} + v_{0})} \cdot 2s = \frac{2s}{v_{1} + v_{0}}$ + +$a_{0} = \sqrt{ \left(\frac{v_{1}^2 - v_{0}^2}{2s}\right)^2 + \left(\frac{v_{0}^2}{R}\right)^2 }$ + +$a_{1} = \sqrt{ \left(\frac{v_{1}^2 - v_{0}^2}{2s}\right)^2 + \left(\frac{v_{1}^2}{R}\right)^2 }$ + +- $v_{0} = 54 \text{ km/h} = 15 \text{ m/s}$ +- $v_{1} = 18 \text{ km/h} = 5 \text{ m/s}$ + +$a_{t} = \frac{5^2 - 15^2}{2 \cdot 800} \text{ m}\cdot\text{s}^{-2} = \frac{25 - 225}{1600} \text{ m}\cdot\text{s}^{-2} = -\frac{200}{1600} \text{ m}\cdot\text{s}^{-2} = -0.125 \text{ m}\cdot\text{s}^{-2}$ +- mínus, takže vektor míří opačným směrem + +### Výsledek + +$T = \frac{800}{5 + 15}\cdot 2s = \frac{1600}{20}s = 80s$ + +$a_{0} = \sqrt{ (-0.125)^2 + \left(\frac{15^2}{800}\right)^2 } \text{ m}\cdot\text{s}^{-2} = 0.308 \text{ m}\cdot\text{s}^{-2}$ + +$a_{1} = \sqrt{ (-0.125)^2 + \left(\frac{5^2}{800}\right)^2 } \text{ m}\cdot\text{s}^{-2} = 0.129 \text{ m}\cdot\text{s}^{-2}$ \ No newline at end of file diff --git a/KFY FYI1/_assets/priklad1.jpg b/KFY FYI1/_assets/priklad1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..9d4523c0e68e67b04d01451bc86eb0d7313e60f6 GIT binary patch literal 17510 zcmeHucU)6j*7gYjLJz$Yx}bC!t00tO64KtOsCP?RPe1O#a!y$Yyw zPpd?&SOA8PY#wH>LlaLS-6Rr*GAHVdEW2PQlE=%Em5uMo9SVIY}vLnai?QR8-Z}H8i!f z4R72uGQMSEYHMfj;OOM+a@WV#?;gTGAT0c0L}b*X=;V~gscGp?o@Ttr%P%M_DlRFl zeqB>rSKsiav7@uAyQlYEU;oJH*!aZcr_WQUMfB2_<*zHNYuKILz5Ro4xbKHQ%LM|! ze-(@H{8zz#BNsslAR=OdN+CbX1tPjf_yf}slki?5rB}WVvG!u%6MsO)cr_`vs-2u) z!T`f$<2_8lEFg&z#QrSWUnKi;f`$H9lKm>!-{qPDsK6k?;(_S^Mc~*lhbV*s5C;D5 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