A power output from a certain experimental car design to be shaped like a cube is proportional to the mass m of the car. The force of air friction on the car is proportional to $A v^{2}$ , where v is the speed of the car and A is the cross-sectional area. On a level surface the car has a maximum speed $V_{\max}$ Assume that all versions of this design have the same density. Then $V_{\max}$ is proportional to $m^{1 / C} .$ Find C.

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answer is 9.

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Detailed Solution

$A v_{\max}^{2} = F = \frac{k m}{v_{\max}} \to 1 (\begin{array}{l} ∵ p = F v_{\max} \\ p \propto m = k m \\ k m = F v_{\max} \\ ∴ F = \frac{k m}{v_{\max}} \end{array})$
$\begin{array}{l} & m \propto v \\ m \propto l^{3} \\ l \propto m^{1 / 3} \\ A \propto l^{2} \end{array}} A \propto m^{2 / 3} \to 2$

$\Rightarrow m^{2 / 3} v_{\max}^{2} \propto \frac{m}{v_{m a x}}$
$\Rightarrow v_{m a x} \propto m^{1 / 9}$
$\frac{1}{c} = \frac{1}{9} \Rightarrow c = 9$