A Jetta and a Corvette travel the same distance from Columbus to Cincinnati. The Jetta travels half of the distance at u miles per hour and the other half at v miles per hour. The Corvette travels half of the time at u miles per hour and the other half at v miles per hour. Which car gets to Cincinnati first?
This problem seems right away to be impossible, because we do not know what the speeds u and v are. Daunting, eh?
From the list of heuristics, #14, make the problem simpler, is a good one to start with. Plugging in specific values for the variables gives good insight as to what is going on. Then, for the proof, we can use #10, write an equation, or simply get a bright idea. The reassuring #18, identify needed information clearly, is hovering in the background — it turns out that the values of u and v are not part of the answer.
Using the Problem with Students
I find that right away students object to not knowing what u and v are, particularly which is faster. That is my cue to say, “Well, let’s try some values and see whether it matters.” This leads us to the pictures and calculations shown below. Along the way, we put off the student who says immediately that the Corvette will arrive first because everybody knows that Corvettes are faster than Jettas. I glare. Or I say maybe the Corvette got stopped and was given a speeding ticket.
Coincidentally, it turns out that the Corvette is faster, as the numerical examples show. But why? This is where somebody needs to come up with a bright idea. I don’t give it away; I let the class worry about it overnight, or even longer. Sometimes I will let a problem like this one hang on for days and days.
I will ask at some later point what happens if u = v. The cars will arrive simultaneously, but it is worth doing the calculations to see why.
Solving the Problem
One approach is simply to assign several arbitrary pairs of values to u and v and see what happens. Let’s start by setting u = 50 mph and v = 100 mph and assume for convenience that the total distance is a nice round 200 miles (which it is not, but no matter).
t/2 50 + t/2 100 = 200
25t + 50t = 200
75t = 200
t = 2 2/3 hours
The Corvette arrives first: 20 minutes ahead of the Jetta.
Now we try it the other way around: Set u=100 mph and v=50 mph.
The mathematics is the same, so the total time is, again, 2 2/3 hours.
So we see that it doesn’t matter which of u or v is faster.
If you try other values for u and v, you will get the same results. (Maybe make a spreadsheet.)
Once we convince ourselves that the Corvette is always going to get there first, we can try to set up some algebra to see why. But this is elusive; it turns out that a very bright idea is all we really need.
And here it is: The more time a car spends going the faster speed, the farther it will get. The car that spends half the time going at the faster speed will get more than halfway there at that speed, and will thus be ahead of the car that went only half the distance at that speed. Furthermore, the car that goes more than halfway at the fast speed will go less than halfway at the slow speed, thus the other car will spend more time going more slowly.