Slowest shortcut Suppose you are standing in a field near a straight section of railroad tracks just as the locomotive of a train passes the point nearest to you, which is mi away. The train, with length mi , is traveling at 20 mi/hr. If you start 4 3 running in a straight line across the field, how slowly can you run and still catch the train? In which direction should you run?
Solution Step 1: Consider that one is standing in a field near a straight section of railroad tracks just as the locomotive of the train passes the point nearest to one, which is miaway. 4 The train has length of 3i, and is travelling at20 mi /hr. If one starts running in a straight line across the field, it is required to know with what speed and in what direction one should run. Step 2: Consider the following figure The parallel lines show the railroad, the train passes point as in one is standing nearest to the railroads that is at point , and the train be initially at point such that the person when runs, manages to catch the train at point even when he runs at the slowest speed. Then 1 AB = 4 (from given) Let the speed of the running man be mi/hr And he should run at angle to the line Such that Then the time taken by the man to reach point is equal to the time taken by the train to reach point From the formula of time: Time taken by man to reach point is given by If the distance is then by cosine formula, the distance is equal to 1 Therefore, time taken by man to reach pointCis given by 4x cos And the time taken by the train to reach the point is given by Equate both the times, that is time taken by man to reach point, and the time taken by train to reach point 1 1 4x cos = 60 4x cos = 60 x = 60 4cos x = 15 cos