When a vehicle is exceeding the posted speed limit of 55 mph, 60, 70, 75, etc, in increments of 10 mph, how long must the vehicle remain at that increased speed to actually reduce the overall drive-time by 1 min, 2 mins, etc?
Are you thinking about some formula -- into which you would plug the total distance, speed limit, the number of 10-mph increases, and the number of minutes by which you would like to decrease the total trip time -- to calculate the number of minutes that you must at drive the increased speed?
Determining such a formula will take some effort; I would first need to know whether this endeavor is frivolous. What motivates your inquiry?
PS: Your GPS gizmo trilateralizes your exact location thousands of times per second using the four (out of 24) satellites in orbital position to accomodate you. Information moves back and forth between you and these satellites at the speed of light. The gizmo knows nothing about speed limits. The gizmo itself calculates your speed based on how far you've moved since the last time it fixed your location. The gizmo's reported information about things like current speed and remaining trip time are generated entirely by software and map-database data within the gizmo.
Here's some more preliminary info.
As your desire is to decrease the total trip time by mere minutes, you should first convert all of your speeds from miles per hour to miles per minute.
EG:
There are 60 minutes in 1 hour
55 miles per hour split into 60 equal pieces yields the equivalent speed of 11/12ths of a mile per minute.
55/60 = 11/12
Distance traveled is the product of constant speed and elapsed time.
d =
r*
t
Converting 10 miles per hour, we get 1/6 mile per minute.
Hence, your slower speed is
r and your faster speed is
r +
n*1/6 (where n = the number of 10-mph increases).
Now, if you drive at speed
r for a total of
t minutes AND you drive at speed
r +
n/6 for a total of
T minutes, then the entire distance may be modeled by:
d =
r*
t + (
r +
n/6)*
T
You seem to be interested in knowing the value of
T from given values for
d,
r, and
n.
A formula may be derived by using some algebraic relationships. (We could also break the trip distance into two segments -- one for each speed.)
d =
d1 +
d2
d1 =
r*
t
d2 = (
r +
n/6)*
T
With some substitutions and manipulations, we might get an equation looking something like the following that may be solved for
T.
t +
T -
x =
d1/
r + (
d -
d1)/(
r +
n/6)
where
x represents the number of minutes by which you desire to reduce the trip time.
Of course, if we were to have a single scenario, things would become much simpler.
EG: you know that your trip distance is 121 miles; you know that the speed limit is 55 mph; you know that you're only willing to increase that speed by one 10-mph increment; and you know that you want to arrive at your destination one minute earlier.
d = 121
r = 55
n = 1
x = 1
These relationships immediately lead to this system of two equations:
11/12*
t1 + 13/12*
t2 = 121
t1 +
t2 = 131
The solution to this system tells us that you must drive at 65 mph for a total of 5.5 minutes, with the remaining 125.5 minutes being driven at 55 mph.
Cheers
