How to Find the Distance Between Two Points
Here we have a basic relative position problem. We have a train and a plane traveling the same route at different speeds and departing at different times, and were asked when will the plane overtake the train? The first thing we want to do is write the equations of motion for both the plane and the train. Let’s start with the plane. The plane departs 1:30 P.M. traveling 500 miles an hour. Our equation of motion for the plane is just going to be the position at time t is equal to 500 -because it’s 500 miles an hour- times t, t is going to be measured in hours starting at 1:30 P.M.
To find the position of the train, we have to do a little back checking. At 1:30 P.M. the train which departed at 7:30 A.M. has already been traveling for six hours. If the train has been traveling for six hours and 50 miles an hour, that means at 1:30 P.M. it’s already gone 300 miles. We’ll start it out with 300 miles plus it’s traveling at 50 miles an hour, so we have 50 times t. To figure out when the plane overtakes the train -another way of saying that is, when are they at the same position? – we’ll set these two equations equal to one another and solve for t.
That will tell us how long after 1:30 it is that the plane overtakes the train. We can write this as 500t is equal to 300 plus 50t. Now we’ll subtract 50t from both sides, that will eliminate this one and that’ll give us 450t is equal to 300, or t equals 300 over 450 which is the same as 2/3. 2/3 of an hour is the time past 1:30 that it takes for the plane to overtake the train. Now, 2/3 of an hour is 40 minutes. We’ll just add 40 minutes to 1:30 P.M., and that gives us 2:10 P.M.