Math CentralQuandaries & Queries


Question from david, a student:

If two cars starts to drive at point 1 and go in a circle when will
they meet?
(one cars speed is faster the other ) all I need is a formula

Hello David,
I will try to help you get started.  Let’s start by forgetting for now that the track is circular, and I will assume that the cars travel in the same direction and at a constant and instantaneous speed (of course life is not like this but we assume this to make the question “easy”).
Let’s imagine we are driving down a straight section of the highway with the cruise control on 110 km/hr.  A car is about to pass us with cruise control set at 125 km/hr (which is a typical situation where I am from).  How do I determine a formula for how far the faster car will be ahead of us for any time, t?
First notice that we do not have to picture both vehicles moving.  Instead we simplify the problem by using an equivalent situation, namely that we are stationary and the other car is travelling at 15 km/hr.

I determine the distance between me and the car by multiplying 15 km/hr by how long it has been since the car passed us (in hours).  So, the distance is given by  d(t) = 15t.

Now let’s return to the circular track.  Both cars are moving around in the same direction at different speeds.  How much faster is the one car?  Imagine the first car to be stationary and the faster car to be moving around at the speed you just determined.  How long will it take to go around the track?  What is the length of the track?

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