## Wednesday, December 30, 2009

### Stopping distance puzzle.

Here is a quick puzzle. It is pretty easy so you might try computing the answer in your head. Suppose you are driving at 60 mph and apply the brakes. Suppose you can brake with a force of 1g. How far will you travel before stopping? For this puzzle assume 1g = 32 ft/(sec)/(sec).

Added note: As discussed in the comments I was a bit sloppy when I said "brake with a force of 1g". What I meant of course was brake so as to produce a deceleration of 1g. See this wikipedia article for much more about this common misnomer.

1. Seems like 121ft, but just in case i made a mistake let us take a safe bet and ask a [ put a name of dem. senator of your choice here ] to please stand at 60ft or less.

2. I got 242 feet, which suggests one of us dropped a factor of two. Most likely me.

And converting from miles per hour to feet per second took this way out of the realm of things I am at all likely to do in my head -- nothing that needed a calculator, but way too many steps to keep straight without paper.

3. Oh, whoops, I never did the 1/2 part of the KE formula. Yeah, 121 feet is right.

4. Wait -- does the problem work? I was blithely thinking that the mass of the car can't matter, because if it did, it'd be given, so I left it at 1; 121 feet is the stopping time for a car massing one slug, if I remember the English unit of mass correctly.

But a heavier car is going to have more KE, and there's nothing about the E=Fd formula that allows us to ignore that. So the answer is 121 feet x (the mass of the car in slugs), if I've got it straight.

5. The mass does not enter the calculations. Although the term "force" may be misleading here - it is really an acceleration (g). Once you know the deceleration, the mass becomes irrelevant (it is the brakes that need to provide the force -- a function of mass -- to deliver the deceleration).

6. Regarding converting 60 miles per hour to feet per second, this is not too difficult. First note 60 mph is 1 mile per minute (as there are 60 minutes in a hour). 1 mile per minute is 5280 feet per minute (as there are 5280 feet in a mile). A minute is 60 seconds so we just have to divide 5280 by 60 to get the equivalent of 60 mph in fps. Conveniently this comes out as exactly 88 fps. Of course it helps if you have done this before and remember the result.

The remainder of the calculation is also easy, we want 88*88/(2*32) = 88*88/(8*8)=11*11=121. Not a difficult problem but I thought it was neat that the answer in feet is an integer.

7. All right, that does make sense -- I got confused by thinking that 'force' meant force, and didn't actually think about the units.

But the integral answer is, in fact, neat -- I'll have to remember this one when my kids get to high school physics.