### Project #1

GOALS:

• To gain experience dealing with unusual problems for which there may not be a single right method of solution.  The answer may take many forms and needs to be explained in detail through the use of words, numbers, formulas, and graphs.
• To gain experience explaining mathematical results in English.
• To practice working with others on a mathematical problem.
• To become familiar with the members of the group with whom you will be working on these projects.

Harold Hadley is very lazy and very rich.  Oh, and he loves to ride his bicycle.  Yes, these may seem contradictory, but there it is.

As Harold rides his bicycle through Salem, he often encounters this problem:  As he rounds a certain bend, he sees a car ahead stopped at a red light.  The car is blocking Harold’s path, so he needs to slow down, but being lazy, he hates to slow down any more than necessary – for then he will have to expend energy to regain his lost speed.

Now the traffic signal timing and Harold’s cycling habits (including speed) are so consistent, that the situation is often precisely this:  His velocity is 20 m.p.h. and he sees the car when his front wheel is 100 feet behind the car’s rear bumper.  Also, these cars seem to accelerate pretty consistently at 7 ft/sec/sec when the light turns green, and the light changes just 3 seconds after Harold sees the car.

You may wonder why Harold doesn’t just get in the habit of coming around this bend at a slower speed, but often the light is green, there is no stopped car, and he need not slow at all.

So Harold is commissioning your consulting team to answer this question:  If he brakes at a constant deceleration starting at the instant that he sees the stopped car, what should that deceleration be so that he doesn’t hit the car but retains as much of his speed as possible?  Oh, and what would happen (give great mathematical detail) if he didn’t brake at all? You should neglect any non-braking deceleration that would occur due to air resistance or other friction.  Granted, these may seem silly questions to invest so much time on, but Harold is rich, and can hire people to do what he wants.  By the way, it’s probably best not to bring up to Harold the vanity of assuming that he can brake at precisely a given deceleration.  Humor him.