Where will two trains meet if train A leaves Chicago traveling at 35 mph and train B leaves Washington D.C. traveling at 45 mph?

Answer 1

When trains meet it's called a train wreck. Let's hope they're not on the same track. But seriously, the question is, well, silly, because we don't know what tracks the trains are on or in what direction they're traveling. If we assume there is a perfectly straight track connecting the two cities and train A is heading for D.C. and train B is heading for the "Windy City," the collision will occur closer to Chicago. How much closer? you ask. The distance covered by train A will be D * 35/(35+45) = 35D/80 = 7D/16, where D is the distance between Chicago and Washington. The distance between the two cities -- as the crow flies -- is 600 miles. So the collision will occur 262.5 miles east of Chicago. Train A is covering 35 miles of track per hour; train B is covering 45 miles per hour. The distance between them, however, is reducing by 35 + 45 = 80 miles per hour. If there is 600 miles of track between them to start out with, it will disappear in 600/80 = 7.5 hours. In that time, train A will have traveled 262.5 miles. Let's use basic linear motion equations to solve the problem. It should be easy because constant velocity is involved; that is, there is no acceleration, so the equations will be simple. The general equation is s = s0 + vt, where s is the distance traveled, s0 is the initial distance, v is the velocity, and t is the time. For our system, let's assume Chicago is the "origin." Let's also assume that traveling east (from Chicago to D.C.) is positive. So the equation for train A is sa = sao + vat = 0 + 35t = 35t (Note that sao is zero because Chicago is the origin.) The equation for train B is sb = sb0 + vbt = 600 + (-45)t = 600 - 45t (Note that vb is negative because train B is traveling west, which is the negative direction.) The collision occurs when sa = sb. So, 35t = 600 - 45t. Adding 45t to both sides yields 80t = 600, so t = 7.5. Plugging that result into the equation for train A, we get sa = 35 * 7.5 = 262.5. Sheesh:

Closing @ 80 mph, the trains will "meet" at 35/80 = 43.75% of the total distance from Chicago.