Opgave_20.pdf

# Opgave_20.pdf - Opgave 20[Naturlig gang Der findes en...

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Opgave 20. [Naturlig gang] Der findes en optimal fart ved hvilken vi bruger mindst energi på at gå. Denne optimale fart 𝑣𝑣 tænkes at kunne afhænge af personens masse 𝑚𝑚 , personens højde samt tyngdeaccelerationen 𝑔𝑔 . a) Bestem ved hjælp af dimensionsanalyse et udtryk for den optimale fart. Kommentér udtrykket. b) Hvor meget hurtigere er den optimale fart for en person der er 1.90 m høj i forhold til en person der er 1.65 m høj? a) Vi vil bestemme 𝑣𝑣 = 𝑓𝑓 ( 𝑚𝑚 , , 𝑔𝑔 ) eller 𝐹𝐹 ( 𝑣𝑣 , 𝑚𝑚 , , 𝑔𝑔 ) = 0 . Vi leder derfor efter en dimensionsløs parameter 𝜋𝜋 = 𝑣𝑣 𝑎𝑎 𝑚𝑚 𝑏𝑏 𝑔𝑔 𝑐𝑐 𝑑𝑑 . Dimensionsmatricen er 𝑣𝑣 𝑚𝑚 𝑔𝑔 M 0 1 0 0 L 1 0 1 1 T 1 0 0 2 Vi ser, at massen er dimensionelt irrelevant så den kan ikke indgå. Vi kan opskrive den reducerede dimensionsmatrix. 𝑣𝑣 𝑔𝑔 L 1 1 1 T 1 0 2 𝑎𝑎 + 𝑐𝑐 + 𝑑𝑑 = 0, −𝑎𝑎 − 2 𝑑𝑑 = 0 To ligninger med tre ubekendte, da farten er den afhængige variabel sætter vi
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