Chapter 3, Problem 42P

This is a symbolic version of Problem 29. A river has a steady speed of v_s. A student swims upstream a distance d and back to the starting point. (a) If the student can swim at a speed of v in still water, how much time t_up does it take the student to swim upstream a distance d? Express the answer in terms of d, v, and v_s. (b) Using the same variables, how much time t_down does it take to swim back downstream to the starting point? (c) Sum the answers found in parts (a) and (b) and show that the time t_a required for the whole trip can be written as

$t_a\text{ = }\frac{\text{2}d/v}{\text{1 - }{{\displaystyle v}}_s^2/{{\displaystyle v}}^2}$

(d) How much time t_b does the trip take in still water?

(e) Which is larger, t_a or t_b? Is it always larger?