+1 862 207 3288 “Each time the Apollo astronauts returned from the moon circa 1970,they took great care to reenter Earth’s atmosphere along a paththat was only a small angle ÃŽ± from the horizontal. This wasnecessary in order to avoid intolerably large ‘g’ forces duringtheir reentry. To appreciate their grounds for concern, consider the idealizedproblem ^2 where K and H are constants and distance s is measured downrangefrom some reference point on the trajectory, as shown in thefigure. This approximate equation pretends that the only force onthe capsule during reentry is air drag. For a bluff body such asthe Apollo, drag is proportional to he square of the speed and tothe local atmospheric density, which falls off exponentially withheight. Intuitively, one might expect that the decelerationpredicted by this model would depend heavily on the constant K(which takes into account the vehicle’s mass, area, etc.); but,remarkably, for capsules entering the atmosphere (at “s =-∞”) with a common speed V[o], the /maximum/deceleration turns out to be independent of K.”
(a) Verify this last assertion by demonstrating that this maximumdeceleration is just V^2[] / (2eH)
(b) Also verify that any such spacecraft at the instant when it isdeceleratng most fiercely will be traveling exactly with speedV[o] / √e , having by then lost almost 40% of itsoriginal velocity.
(c) Using the plausible data V[o] = 11 km/sec and H= 10 /(sin(ÃŽ±)) km, estimate how small ÃŽ± had to be chosen so asto inconvenience the returning travelers with no more than 10g’s.
