3. An object of mass m is dropped into the sea at the surface. It is...

 

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3. An object of mass m is dropped into the sea at the surface. It is subject to a constant gravitational acceleration g and water resistance. (a) (4 points) If the water resistance is proportional to the object's velocity v(t) with coeffi- cient 60, so that Newton's second law gives dv 9- dt - b m show that the object's velocity as a function of time t is given by v(t) = mg b (1-e-bt/m). We take downward to be the positive direction and the object is at rest when it is dropped at t = 0. (b) (4 points) If instead the water resistance is quadratic in the velocity, so that Newton's second law gives dv dt - 9 -v m dv dt = find the object's velocity as a function of time t, again assuming that the object is dropped from rest. [Hint: Write Newtons' law as 9 (1-k) where k = b/mg. Then the algebra is a little less messy. After obtaining the final answer in terms of k, remember to replace k by b/mg.] (c) (2 points) If the object could fall forever under these conditions without hitting the bot- tom, show that as to the velocity in part (a) approaches terminal velocity vmg/b. What terminal velocity does the velocity in part (b) approach? 3. An object of mass m is dropped into the sea at the surface. It is subject to a constant gravitational acceleration g and water resistance. (a) (4 points) If the water resistance is proportional to the object's velocity v(t) with coeffi- cient 60, so that Newton's second law gives dv 9- dt - b m show that the object's velocity as a function of time t is given by v(t) = mg b (1-e-bt/m). We take downward to be the positive direction and the object is at rest when it is dropped at t = 0. (b) (4 points) If instead the water resistance is quadratic in the velocity, so that Newton's second law gives dv dt - 9 -v m dv dt = find the object's velocity as a function of time t, again assuming that the object is dropped from rest. [Hint: Write Newtons' law as 9 (1-k) where k = b/mg. Then the algebra is a little less messy. After obtaining the final answer in terms of k, remember to replace k by b/mg.] (c) (2 points) If the object could fall forever under these conditions without hitting the bot- tom, show that as to the velocity in part (a) approaches terminal velocity vmg/b. What terminal velocity does the velocity in part (b) approach?