What Goes Up . . . (a) It is well known that the model in which air resistance is ignored, part (a) of Problem 36, predicts that the time ta it takes the cannonball to attain its maximum height is the same as the time td it takes the cannonball to fall from the maximum height to the ground. Moreover, the magnitude of the impact velocity vi will be the same as the initial velocity v0 of the cannonball. Verify both of these results.
(b) Then, using the model in Problem 37 that takes air resistance into account, compare the value of ta with td and the value of the magnitude of vi with v0. A root-finding application of a CAS (or graphic calculator) may be useful here.
Reference: Problem 37
How High?—Linear Air Resistance Repeat Problem 36, but this time assume that air resistance is proportional to instantaneous velocity. It stands to reason that the maximum height attained by the cannonball must be less than that in part (b) of Problem 36. Show this by supposing that the constant of proportionality is k = 0.0025. [Hint: Slightly modify the DE in Problem 35.]
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