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 drag coefficient is k = 0.0025. [Hint: Slightly modify the DE in Problem 35.]
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How High?—No Air Resistance Suppose a small cannonball weighing 16 lb is shot vertically upward with an initial velocity v0 = 300 ft/s. The answer to the question, “How high does the cannonball go?” depends on whether we take air resistance into account.
(a) Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by d 2s/dt 2 = −g (equation (12) of Section 1.3). Since ds/dt=v(t) the last differential equation is the same as dv/dt = −g, where we take g = 32 ft /s2. Find the velocity v(t) of the cannonball at time t.
(b) Use the result obtained in part (a) to determine the height s(t) of the cannonball measured from ground level. Find the maximum height attained by the cannonball.
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