Iterative Tangents

A curiosity journal of math, physics, programming, astronomy, and more.

Math

Saving time by speeding

For a couple years I had a 45-minute commute, mostly by freeway, and it made me wonder how much time I could I save by speeding. Typically I stay pretty close to the speed limit since I've never found it that difficult either to leave early enough or to accept being late, and I derive a lot of mental ease from not being in a hurry and not having to worry about getting caught speeding. But I tend to do a lot of mental math on solitary, low-stress drives, and how much time I could save by speeding was some of it.

Rate times time equals distance, so given two speeds and the same distance, we have two different times:

upper R 1 t 1 equals d equals upper R 2 t 2 upper R 1 t 1 equals upper R 2 t 2

To find the time saved by driving faster, solve for t2:

StartLayout 1st Row 1st Column t 2 2nd Column equals StartFraction upper R 1 t 1 Over upper R 2 EndFraction 2nd Row 1st Column Blank 2nd Column equals StartFraction upper R 1 Over upper R 2 EndFraction t 1 EndLayout

Most of the southern part of Michigan's lower peninsula has speed limits of 70 miles per hour. On a 45-minute commute at 70 mph, driving 75 mph instead would give me a commute of

StartLayout 1st Row 1st Column t 2 2nd Column equals StartFraction 70 normal m normal p normal h Over 75 normal m normal p normal h EndFraction 45 normal m normal i normal n 2nd Row 1st Column Blank 2nd Column equals StartFraction 14 Over 15 EndFraction 45 normal m normal i normal n 3rd Row 1st Column Blank 2nd Column equals 42 normal m normal i normal n EndLayout

To me, 75 mph feels a lot faster than 70 mph, so I was surprised to realize that over 45 minutes it would only save me three minutes of travel time.

Of course, on a longer trip those small amounts of time add up. I've driven Canada's 401 to New England several times, and the speed limit is 100 kilometers per hour, though almost everyone seems to drive 120 kph. On a 14-hour drive,

StartLayout 1st Row 1st Column t 2 2nd Column equals StartFraction 100 normal k normal p normal h Over 120 normal k normal p normal h EndFraction 14 normal h normal r 2nd Row 1st Column Blank 2nd Column equals five sixths 14 normal h normal r 3rd Row 1st Column Blank 2nd Column equals 11 normal h normal r 40 normal m normal i normal n EndLayout

On a 14-hour drive, a tired driver may be more dangerous than a slightly faster driver, so saving two hours by speeding could be the safer option on long drives, but good luck explaining your way out of a ticket with that reasoning.

These days I more commonly drive about 40 minutes on mostly 55-mph country roads. Occasionally someone will pass me going 60 or 65. Given the length of my drive (and probably theirs), speeding just doesn't seem worth it. Here's a chart of the time speeding saves on different length drives:

Speed (mph)10 min15 min20 min25 min30 min35 min40 min45 min50 min55 min60 min
5610s16s21s26s32s37s42s48s53s58s1m
5721s31s42s52s1m1m1m1m1m1m2m
5831s46s1m1m1m1m2m2m2m2m3m
5940s1m1m1m2m2m2m3m3m3m4m
6049s1m1m2m2m2m3m3m4m4m5m
6159s1m1m2m2m3m3m4m4m5m5m
621m1m2m2m3m3m4m5m5m6m6m
631m1m2m3m3m4m5m5m6m6m7m
641m2m2m3m4m4m5m6m7m7m8m
651m2m3m3m4m5m6m6m7m8m9m

The same math applies if you're headed to an appointment and realize with, say, 10 minutes to go that you're going to be a couple minutes late. Driving 10 mph faster at that point is only going to get you there one minute earlier.

We'll complicate this simplistic model in the next post.