Round trips with and against a wind.

An airplane makes a round trip where the one-way distance is 1,000km. On the out-leg the plane faces a head-wind of 50km/h. While on the return there is a tailwind of 50km/h. If the speed of the plane in still air is 400km/h. What is the total time for the trip?

a. A qualitative argument. Before you solve the problem, think about it in a “qualitative” way: Sketch a rough graph of a function giving the total time for the round trip in terms of the wind speed as the wind speed varies from 0 to 400kph. Compared with the total time for a round trip with no wind, do you think the time for the round trip with the wind is (i) less, (ii) the same or (iii) more?

b. A numerical answer. Answer the question of the problem. Does your answer support your qualitative response in part a.

c. A general answer. The numerical answer does not reveal much about the structure of the situation. Solve the problem again, this time expressing the total time in terms of general parameters for the total distance, the air speed of the plane, and the wind speed. There are many different equivalent symbolic expressions that will express the total time. Try to “coax” the expression you arrive at into a simple form.

d. The general answer refined. Express the total time with no wind (call it t_{0}) in terms of the given parameters. Use t_{0 }to get a more revealing expression for the total time with wind. There is a connection of this problem to special relativity through a "Lorenz Transformation." What is the connection?

e. The motion functions. Functions have not played a role so far in the analysis. Give an alternatie approach by modeling the situation with the motion function of the plane's outbound and return trip. Graph these functions.

f. The dimensionless factor. A dimensionless factor 1/(1-r^{2}), where r is the ratio of the wind speed to the plane's speed, appears in the expression for the total points found in parts c & d. Analyze this factor as a function of r, and graph this function.