A Velocity-Time Problem

Date: 02/16/99 at 11:22:53
From: Sarah
Subject: Math

This is a question from my driver's education class, but it really is a 
mathematical problem: 

If you drive 40 mph instead of 30 mph, you save 30 seconds per mile.  
But if you make the same 10 mph increase from 70 mph to 80 mph, you 
only save 5.6 seconds per mile. Why is this?

I really do not know how to solve this question. I found out the 
formula for how they figured out how many seconds you save, but that 
does not seem to help. 

Date: 02/18/99 at 03:06:46
From: Doctor Mitteldorf
Subject: Re: Math

This really is a question about how math works in the real world, and 
when you answer it, you will have a deeper understanding of one piece 
of that world. I suggest you start by calculating those numbers again.  
How much time does it take to go a mile at 30 MPH? at 40? How much do 
you save? And again, how much time does it take to go a mile at 70 MPH 
and 80 MPH?

You can ask the question in another way: How much extra distance do you 
go in a minute when you increase speed from 30 to 40? and from 70 to 
80? And here is another way to look at it: Suppose the speed of a car 
was measured in "seconds per mile" instead of in "miles per hour."  
60 would still be 60, but what would 70 and 80 be? How about 30 and 40 
and 50?

Now, in this world where people's speedometers read seconds-per-mile, 
if you slow down from 40 to 50 seconds per mile, how many seconds do 
you lose per mile? And how many miles per hour do you lose? Try 
answering the same question when you slow from 60 to 70 seconds per 
mile.

Please write back and let me know what answers you get, and whether 
they give you any ideas about why and how things come out in this 
strange way.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/