Real-Life Math
You are about to enter a professional downhill race. The course,
as mountain bike racer Duane Nickull describes one of his courses, "is approximately
4 to 5 minutes long and will have a mixture of single track, wide-open fire
roads, jumps, corners and just about everything else a person can dream up."
You
have mastered the basic components of riding a mountain bike down a course,
and you are just as capable as everyone else in the race. When you are at
this level, Nickull says, it's time to "concentrate on 2 main areas:
bike setup and the actual lines you take around a corner to look for lower
elapsed times.
"What a person really has to do is become an expert
in physics -- which, of course, involves lots of math."
The racetrack
is 35 percent corners, 20 percent corner exits and 45 percent straight (where
you would accelerate) and is 5 kilometers long.
If you use a heavier
tire, it will corner better and have a higher speed on the exit of the turn.
You estimate that your speed on the corners with the heavier tire is 20 km/hr
and that your speed on the exit is 25 km/hr. However, the heavier tires make
it harder to accelerate to full speed quickly, so your average speed on the
straight is 40 km/hr.
If you put on a lighter tire, you will exit the
corner with a speed of 5 km/hr less than the heavier tire. You estimate your
corner speed will be 6 km/hr less than the heavier tire. The lighter tires
do better on the straight areas, allowing you to reach an average speed of
45 km/hr on the straight sections.
Using the necessary information,
calculate which tire will result in the lowest elapsed time.