So I just now made the ~200 mile drive from Austin (my current residence) to Dallas (where I grew up), both Texas of course, to take care of some stuff. I’ll be driving back tonight, wee. I have to say, that particular drive is one of the dullest in existence. It’s not particularly long, traffic is normal, and nothing special per say, there’s just nothing to look at the whole way, And large gaps of road with no stops in between. At least in the desert or middle states you have a little variety or mountains hopefully to look at. I’ve made 24+ hour straight trips back and forth from Canada that I’ve loathed less :-).
Anywho, whenever I’m on a car trip of more than 100 miles, my mind always turns to counting down miles and running simple arithmetic in my head to calculate how much longer it will take at my current speed to reach my destination, how much time I can cut off if I went faster, etc. This time around my mind turned towards deriving some formulas. This is not the first time this has happened either XD. I have to occupy myself with something when there’s just music to listen to and nothing else to do! Driving is basically a muscle reflex for me on these long drives.
So there are 2 formulas that are useful for this situation.
#1 How much faster you are traveling per minute at different speeds.
#2 How much time you will save at different speeds.
Variables:
H=Higher Speed In MPH
L=Lower Speed In MPH
M=Number of miles to travel
The following are basic proofs of how the formulas work. God... I swore after I got out of geometry I’d never think about proofs again.
The first one is very simple.
Number of extra miles traveled per hour = (H-L)
Number of extra miles traveled per minute = (H-L) mph / 60 minutes
So, for example, if you increase your speed from 20 to 30, you are going 10 miles an hour faster, which is 1/6 of a mile a minute.
The second one is slightly more difficult but much more useful.
h = Time it takes in hours to travel M at H = M miles / H mph
l = Time it takes in hours to travel M at L = M miles / L mph
Difference of time it takes between 2 speeds in hours = h-l
(M/H)-(M/L) [Substituting variables]
(MH-ML)/(HL) [Getting a common denominator]
M*(H-L)/(HL) [Distributive property]
So we can see that time saved, in hours, per mile is (H-L)/(H*L). Just multiply that by M to get total time saved in hours.
With this second formula, we can see that in the higher speeds you go, the difference between the two speeds increase geometrically to get the same type of time savings (because H*L is a divisor, making it inversely proportional).
For example:
If H=20 mph and L=10mph
Time saved = (20-10)/(20*10) = 10/200 = 1/20 of an hour saved per mile, or 3 minutes
If H=30 mph and L=20mph
Time saved = (30-20)/(30*20) = 10/600 = 1/60 of an hour saved per mile, or 1 minute
If you wanted to save 3 minutes per mile when starting at 15 mph...
(x-15)/(15x)=1/20
x-15=15x/20
-15=15x/20-x
-15=-1/4x
x=60 miles per hour
If you wanted to save 3 minutes per mile when starting at 20 mph...
(x-20)/(20x)=1/20
-20=20x/20-x
-20=0 ...
Wait, what? ... oh right, you cant save 3 minutes when it only takes 3 minutes per mile @ 20 mph, hehe.
And if you wanted to save 6 minutes starting at 20mph, you would have to go -20mph, which is kind of theoretically possible since physics has negative velocities... just not negative speeds >.>. I’m sure all it would take is one point twenty one jiggawatts to achieve.
If you’ve actually read this far without getting bored, I congratulate you :-).
Even more sad is the last Dallas-Austin drive I made in which I couldn’t remember the compound continually interest formula and spent a good chunk of the time deriving it in my head (all I could remember was the needed variables
“pert” - principle, e (~2.71 - exp), rate, time).
