Question

Stunt pilots and fighter pilots who fly at high speeds in a downward-curving arc may experience a “red out,” in which blood is forced upward into the flier’s head, potentially swelling or breaking capillaries in the eyes and leading to a reddening of vision and even loss of consciousness. This effect can occur when the non-gravitational part of the centripetal acceleration exceeds 2.5 g’s.

For a stunt plane flying at a speed of 250 km/h, what is the minimum radius of the downward curve a pilot can achieve without experiencing a red out at the top of the arc? (Hint: Remember that gravity provides part of the centripetal acceleration at the top of the arc; it’s the acceleration required in excess of gravity that causes this problem.)
Express your answer to two significant figures and include the appropriate units.

Answer 1

$$ \begin{aligned} &\text { centripetal force at top }=\mathrm{mv}^{2}  / \mathrm{r}+\mathrm{mg}\\ &\begin{array}{l} \mathrm{mv}^{2} / \mathrm{r}+\mathrm{mg}<2.5 \mathrm{mg} \\ \mathrm{v}^{2} / \mathrm{r}+\mathrm{g}<_{-} 2.5 \mathrm{~g} \\ 4822.53 / \mathrm{r}+\mathrm{g}<-2.5 \mathrm{~g} \\ 4822.53 / \mathrm{r}<-1.5 \mathrm{~g} \\ 4822.53 / \mathrm{r}<_{-} 14.7 \\ \text { so } \mathrm{r}, \mathrm{min}=328.06 \mathrm{~m} \end{array} \end{aligned} $$