How long does it take?

Let's say you're in a car, some distance away from your home.  You change your speed so that once you're x miles away, you travel at x mph.  How long will it take to get home, approximately?  Consider two specific cases:

1) You change your speed continiously, so that at, say, 7.5 miles away, you're traveling at 7.5 miles per hour.

2) You drop your speed by one mile an hour for every mile traveled.  That is, at 7 miles away, you go to 7 mph, holding that speed for a mile, then going down to 6 mph at a distance of 6 miles away.

Does it matter how far away you are when you start?

>>6

It's two different cases.

The answer is that it takes ~1.7 times your time metric for case two.  (Again, you'll never get there in case 1, because doing the math means evaluating the logarithm of zero.)  If you're measuring in miles per hour, it'll take about 1.7 hours.  If you're measuring in meters per minute, it'll take about 1.7 minutes to get home. 

Does it matter where you start?  Barely.  If you start 100 miles away, or 50, or 500, it'll still take about the same amount of time, 1.7 hours.  The question is, why?  (If you're familiar with Taylor Series, the answer is obvious.)