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C.) Given that the boat was halfway to Station One before you reversed it, that it started from rest at Base Camp, and that the fan was accelerating the boat forward until the instant you reversed it, where will the boat be when it (instantaneously) comes to rest after you have reversed the fan?
Model: To correctly answer this part, you will have to model the boat's motion starting from Base Camp. The model is still One-Dimensional Motion With Constant Acceleration, but it has to be applied twice. First, there is the eastward acceleration from the time that you leave Base Camp until you reverse the fan. Then, there is the westward acceleration model of (A.).
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It is an extremely common and extremely useful technique to break up a complicated motion into more-easily-modeled pieces. |
Application: This time we are asked for a quantitative answer. You can solve this with equations, but you would have to make up numbers. A better way to do this is to use graphs to understand the symmetry of the equations. Here is an extension of the velocity graph backward in time which shows the acceleration from the forward thrust of the fan as you started out from Base Camp. We know that the slope should simply be the opposite of the slope when the fan is reversed. Further, we assume that you started at rest. Thus, the red part of the graph is a reasonable representation of your motion with the fan accelerating you toward Station One. Notice the obvious symmetry with the part of the graph from (B.) that has been colored blue in the new graph. It is clear that the area under these lines is the same. Thus, using what we know about the relationship between velocity and position, we conclude that the distances traveled are the same. Since the red part corresponds to traveling halfway to Station One, the blue part corresponds to the same distance. The boat has made it to Station One just as it comes to rest! (Note that the blue part terminates at zero velocity.)
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