The relationship between distance traveled, the rate at which we are going, and the amount of time spent traveling can be expressed in one of the most simple and succinct equations: D = R*T. Distance is equal to rate times time. While a simple little equation, we encounter each of these quantitates in our daily lives in different ways. We want distances to be smaller, rates to be ever increasing, and time of travel to vanish towards zero. Yet this relationship remains, ever mutable to our desires and goals.
In traveling to our local grocery store from our home, we generally know the distance. It is fixed and cannot change in that moment however we want it to. Sure, we could move to a home closer to our preferred grocery store, but that is a bit of an ask on a Sunday afternoon. Instead we consider the other two variables, rate and time. It would be nice if the time to get to the grocery store was near zero. But in order to do that our rate would have to be astronomically high. Like most things in life there is a trade off; we can shorten the time traveled, we would just have to travel at unsafe speeds. But is that really the case? Do we really save that much time by going faster?
Example 1
Let's say you live 10 miles from your grocery store. Just based on this information we can make an educated guess as to how long it should take you to get there traveling at different rates.
At 20 miles an hour it would take you 30 minutes to get there.
At 30 miles an hour it would take you 20 minutes to get there.
At 40 miles an hour it would take you 15 minutes to get there.
At 50 miles an hour it would take you 12 minutes to get there.
At 60 miles an hour it would take you 10 minutes to get there.
It is true that as we increase our rate of travel we get to our grocery store in less and less time. But what happens if we double our speed? From 20 miles an hour to 40 miles an hour sure we get there in half the time, from a 30 minute trip to a 15 minute trip, a difference of 15 minutes. Yet from 30 miles an hour to 60 miles an hour we also halve our time, from 20 minutes to 10 minutes, a difference of 10 minutes.
These differences in doubling our speed are 'big' in a sense, we are halving our time. But the amount of time we save, called absolute differences, are increasingly small.
Example 2
It is 3:20 PM and you have to travel 10 miles on a highway to an appointment at 3:30 PM. You enter your destination into your map app and at 60 miles an hour it will take you 13 minutes to get there; 10 minutes on the highway, and 3 minutes with traffic and getting on and off the highway. You hope you can beat that estimate by going a little faster on the highway.
At 60 miles an it would take you 10 minutes to travel the 10 miles, plus 3 minutes: Arrival Time 3:33 PM.
At 65 miles an it would take you 9.23 minutes to travel the 10 miles, plus 3 minutes: Arrival Time 3:32 PM.
At 70 miles an it would take you 8.57 minutes to travel the 10 miles, plus 3 minutes: Arrival Time 3:31 PM.
At 75 miles an it would take you 8 minutes to travel the 10 miles, plus 3 minutes: Arrival Time 3:31 PM.
At 80 miles an it would take you 7.5 minutes to travel the 10 miles, plus 3 minutes: Arrival Time 3:30 PM.
So you technically could reach your appointment on time, but you're traveling 80 miles an hour on the highway. This is legal in very few areas of the country, and increases the risk of an accident and of serious fatalities if there was one, yours or someone else's.
Note how much time you save for each 5 mile per hour increase: less than 50 seconds from 60 to 65 mph, less than 45 seconds from 65 to 70 mph, less than 34 seconds from 70 to 75 mph, and less than 30 seconds from 75 to 80 mph. While your time traveled is certainly decreasing, you aren't saving as much time going from 60 to 70 mph, than you are going from 70 to 80 mph.
So what is going on here? Well we're not actually using the equation D = R*T this way, we're varying R, keeping D the same, and finding out what T is. Mathematically we are using an equivalent form of this equation T = D/R: the time traveled is equal to the distance traveled, divided by the rate of travel. Once we keep D the same, or fixed at 10 miles, we can let these two variables vary, and get the following graph.
This is not a simple relationship. As the rate of travel increases we certainly see that the time decreases, but at ever slower and slower rates. Looking at when our speed is 50 miles an hour (on the horizontal axis) we see the time is 0.2 hours, or 20 minutes. But at 80 miles an hour, our time becomes 0.125 hours, or 7.5 minutes. For each 1 mile an hour we increase our speed we decrease our time, but at a slower and slower rate.
This relationship is called an inverse relationship, meaning as one quantity increases, the other quantity decreases. Looking at this relationship as a function, this would be the rational function f(t) = 10/t. This kind of function has unique properties: as x increases grows forever, the y values could decrease but never actually reach 0.
This is the situation of our relationship T = D/R, as our speed increases the time decreases. But after increasing our speed for a while, the gains we make in time saved are minimal. So where do we stop? Well, we should probably stay at the speed we are comfortable at, that reduces the chances that we get pulled over, harm others, or reduce fatalities. This is how speed limits are usually calculated, to reduce the chances of severe accidents.
Working with mathematical relationships, we sometimes come to a point where we can't make a 'best' decision, called optimization, but we have to decide when something is 'enough'. This usually rests on assumptions about the equation, or limits to what we can do in the real world. We like to think of mathematics in terms of unconstrained solutions, where any solution could viably solve a given equation. Yet we actually live in a pretty constrained world. Traveling 100 mph on the highway is possible, but not practical for a variety of reasons: risk to yourself, risk to others, getting a speeding ticket, etc.
The next time you find yourself trying to beat your map app's estimate, consider what constraints you are working with and how you can adjust them for your next trip. Could leaving a bit earlier give you the time you needed to get there when you want to? Could leaving even earlier allow you to arrive cool, calm, and collected? While we can control the rate of our speed, R, we also have a lot more control over when we leave.
