There and Back Again
Six hundred seventy one miles is a long way to drive in a day. I know that seems pretty obvious, but it is the kind of drive that seems longer once you are an hour in to it, when the "hundreds of miles left" digit hasn't changed. That kind of drive gets you to thinking. About speeding.
It was on a drive like this that I started thinking about the speed I was driving and what minor changes would mean. I know there are a lot of attitudes about speeding, based on who you are, where you are from, and how much control you feel over your machine. I think I fall into a group with a large number of people who are comfortable at 5-10 mph over the limit but uncomfortable in the 15-20 range. Thats not to say I don't enjoy the occasional ride in my friend's Tesla. Or singing along when my pile shakes as I hit 80 on the open road. I do. I just don't think of driving that way most of the time.
But back to my long drive and thoughts on speed. Speed is funny. We normally use units of miles per hour or meters per second or other various distance over time measures. But in practice this time-in-the-denominator thinking is made burdensome by one small truth. We don't say:
"I'll meet you in Tacoma. Its a half hour away, so I'll see you in 30 miles."
Correctly, we say:
"I'll meet you in Tacoma. It is 30 miles away, so I'll see you in a half hour."
We acknowledge the independence of distance and dependence of time, yet give them opposite roles in our measurement. (And we wonder why people have trouble determining which train gets to Chicago first!) All this is made uniquely interesting by freeway speeds that are generally close to 60 miles / h, making the super easy conversion of 1 min / mile. If you go from Seattle (I-5 exit 163) to Tacoma (I-5 exit 133), you don't need the internets to tell you that the 30 miles between those exits should take about 30 minutes (just to know that it is 2.5 h in current traffic). But when you diverge from that easily invertible speed, things get crazy. Twenty mph over (80) saves you 7.5 min, but 20 mph under (40) loses you 15 min. Its a kind of beautiful asymmetry in a hope-you-don't-get-stuck-in-Tacoma sort of way.
The big prize here would be to cut 5 hours off of the trip by going 125 MPH the whole time. I'm not sure my car can get to 125, let alone maintain it if I hit a hill. So, second prize? What about saving nearly an hour and a half by going 10 MPH over the limit (which my friend from NJ says they aren't even allowed to pull you over for). That seems doable!
But what does it get us? Depending on how much you value your time, maybe not much. That hour and a half could be made up by eating lunch in the car and skipping the stop. It could be made up by leaving a bit early. It could be made up by not leaving late.
And the downside case? Stuck in light traffic the whole time and losing 5 hours as my speed drops to 45 MPH. So to summarize, this long trip had a small achievable upside, and large achievable downside. What happens when the distance changes? Can we keep any of that time we saved by speeding, and perhaps fit in one extra activity before starting our trip? I know its been a while, so lets dust off that old spreadsheet and start modeling!
First let us come up with a list of activities:
Time (min) Activity
0.25 Tie your shoes.
0.33 Tie your shoes (double knotted).
1 Clap your hands 802 times (expert).
2 Find your wallet (expert).
5 Clap your hands 802 times (novice).
10 Find your wallet (novice).
or Clap your hands 802 times (beginner).
22 Watch an episode of Colbert Report (no commercials).
30 Watch an episode of Colbert Report (w/ commercials).
60 Watch an episode of 60 Minutes.
Enter the times in cells A4:A12 of the new sheet. Convert to hours by making cell B4 [=A4/60] (entering the portion [in brackets]), and complete the column (either dragging down or double clicking the bottom right corner).
Type the reference speed in cell A1. Lets start with 65 MPH. Next, fill in cell C4 with [=-C$3/($B4-C$3/$A$1)] which is an algebraic way of solving for the speed at which you can make up that much time in that much distance (vs. your reference speed). Complete the column. Complete the row. (The $ should take care of keeping fixed values in place but I usually spot check. The form should still follow [= - distance / (time - distance / RefSpeed)] in all cells). Immediately you might notice that output is somewhat nonsensical. There are negative speeds in there! Those occur because of the physical impossibility of making up more time than the drive would take. You can't make up an hour on a 1 h drive, because you'd be going infinitely fast. You can't make up 2 h either (hence the negative speed).
One way to get this out of the way is to do an "if" gate, where a negative value returns a ridiculously fast speed, of, say 1000 MPH. This value will never appear on our graphs, but gives them the appearance of the infinite asymptote they are approaching. Copy cells B4:B12 and paste special "values" in B14. For cell C14 enter [=IF(C4<0,1000,C4)] and complete the column. Complete the row. Yay! Graph that!
Looks like at freeway speeds it is tough to make up time. Anything under 10 miles and you need some major speeding (the kind that can get you pulled over even in NJ) to make up just 2 minutes. Alternatively, drives over 100 miles lead to nice time gains from your speeding, such that you might even catch up on your TV watching.
Things get a bit more interesting on surface streets though. Bump the reference speed down to 30.
The lower speeds here make the curves look much sharper. Driving ten miles at our new reference speed would take about 20 minutes if we hit all green lights, but to shave even 5 minutes off that would require us to increase our speed by a third. On the other hand, that same level of speeding can save an hour when you go over a hundred miles.
Actually, now that I mention it, stoplights are a much better reason to be late. Forget that double-knotting of your shoe... hitting just one stoplight could put you a minute or two behind schedule. I wanted to see how much time you might lose to stoplights, so I played around with modeling traffic light timing in Excel before realizing:
A) It's hard.
B) It depends on what kind of pattern you have, and
C) It's been done before. Hundreds of thousands of times.
A much easier path is to make some assumptions:
Assumption 1. The number of lights you pass through increases linearly with distance you drive. (If you drive 10 miles you will pass through twice as many intersections as a 5 mile drive.)
Assumption 2. For every X lights you pass through, you will be stopped N number of minutes. (I'll choose 4 intersections, stopping for 1 minute. I don't know which intersections, just total stop time.)
Assumption 3. The spacing for intersections is similar to that in Seattle. Every 5 blocks = major intersection = quarter mile.
This results in something quite interesting. There is no effect of speed on the amount of time you are stopped at lights. Of course this is a little simplistic, and does not account for running yellow lights (which would favor speeding) or timed lights for a given stretch (which discourages speeding), but instead lets us calculate a new ratio of "stoplight time / mile". With the conditions I mentioned in assumption 2, that ratio would be 1 min/mile. Glancing back up at the 30 MPH chart, I can see that to save 1 min on 1 mile (or 10 min on 10 miles etc.) would be... 60 MPH?
Let me double check that. Ten miles at 30 MPH is 20 min. Adding 1 min/mile of stoplight time would bring the total to 30 min. If we tried to get back down to 20 min total trip time we'd subtract 10 min for sitting at stoplights and try to make the 10 miles in the remaining 10 minutes. Yup, thats 60 MPH.
And the sad thing is that I was trying to be conservative with my estimate of stoplight time. Downtown it could be much higher, and remember, this stoppage is distance independent, so we can't just drive 1000 mies to get efficiency (although after we leave the city the stoplight time/mile should drop significantly). And here's a spooky thought (my hedge against whether or not I get in a Halloween post), what if the stoplight time gets to 2 min/mile?
That time is unrecoverable.
But back to my long drive and thoughts on speed. Speed is funny. We normally use units of miles per hour or meters per second or other various distance over time measures. But in practice this time-in-the-denominator thinking is made burdensome by one small truth. We don't say:
"I'll meet you in Tacoma. Its a half hour away, so I'll see you in 30 miles."
Correctly, we say:
"I'll meet you in Tacoma. It is 30 miles away, so I'll see you in a half hour."
We acknowledge the independence of distance and dependence of time, yet give them opposite roles in our measurement. (And we wonder why people have trouble determining which train gets to Chicago first!) All this is made uniquely interesting by freeway speeds that are generally close to 60 miles / h, making the super easy conversion of 1 min / mile. If you go from Seattle (I-5 exit 163) to Tacoma (I-5 exit 133), you don't need the internets to tell you that the 30 miles between those exits should take about 30 minutes (just to know that it is 2.5 h in current traffic). But when you diverge from that easily invertible speed, things get crazy. Twenty mph over (80) saves you 7.5 min, but 20 mph under (40) loses you 15 min. Its a kind of beautiful asymmetry in a hope-you-don't-get-stuck-in-Tacoma sort of way.
But Does It Make Sense?
So really, back to those six hundred seventy one miles. At 65 MPH the inverted speed is 0.923 min / mile. Can I save any time by speeding? Is it going to kill me to go slower than the posted speed? The curve is going to look the same, but bigger.The big prize here would be to cut 5 hours off of the trip by going 125 MPH the whole time. I'm not sure my car can get to 125, let alone maintain it if I hit a hill. So, second prize? What about saving nearly an hour and a half by going 10 MPH over the limit (which my friend from NJ says they aren't even allowed to pull you over for). That seems doable!
But what does it get us? Depending on how much you value your time, maybe not much. That hour and a half could be made up by eating lunch in the car and skipping the stop. It could be made up by leaving a bit early. It could be made up by not leaving late.
And the downside case? Stuck in light traffic the whole time and losing 5 hours as my speed drops to 45 MPH. So to summarize, this long trip had a small achievable upside, and large achievable downside. What happens when the distance changes? Can we keep any of that time we saved by speeding, and perhaps fit in one extra activity before starting our trip? I know its been a while, so lets dust off that old spreadsheet and start modeling!
First let us come up with a list of activities:
Time (min) Activity
0.25 Tie your shoes.
0.33 Tie your shoes (double knotted).
1 Clap your hands 802 times (expert).
2 Find your wallet (expert).
5 Clap your hands 802 times (novice).
10 Find your wallet (novice).
or Clap your hands 802 times (beginner).
22 Watch an episode of Colbert Report (no commercials).
30 Watch an episode of Colbert Report (w/ commercials).
60 Watch an episode of 60 Minutes.
Enter the times in cells A4:A12 of the new sheet. Convert to hours by making cell B4 [=A4/60] (entering the portion [in brackets]), and complete the column (either dragging down or double clicking the bottom right corner).
Don't Forget to Log Out!
Next we need to think about our output. I want to know if I can still make up time given various distances. But 671 miles is very different from 1 mile, so chances are I'll want to use a logarithmic axis to display the data. Since I realize that now, I can choose distances to make the graph look pretty. This can be generalized to any log graph, using the range and number of data points you want. Enter the range in A1. For me that will be [=1000-1] because I'll have the log axis go from 1 to 1000. (Don't try to make it go to zero or you will have problems.) Now decide how many data points you want and put them in the log base (or denominator of an exponent). In my case I'll choose 16 so it looks like [=A1^(1/16)] = about 1.5. This is the factor that each distance will be increased by (basically counting in "one-point-fiveary" instead of decimal or binary). Make C1:S1 the numbers 1, 2, 3, 4...16, and make C2 [=1.5^C1] then complete the row. Copy and paste special "values" into row C3:S3 then delete all that stuff you needed to get your values. In my case I have the values 1.5, 2.25, 3.375... 985.261253. These values aren't nice and round, but on the graph they will be perfectly spaced on the log axis.Type the reference speed in cell A1. Lets start with 65 MPH. Next, fill in cell C4 with [=-C$3/($B4-C$3/$A$1)] which is an algebraic way of solving for the speed at which you can make up that much time in that much distance (vs. your reference speed). Complete the column. Complete the row. (The $ should take care of keeping fixed values in place but I usually spot check. The form should still follow [= - distance / (time - distance / RefSpeed)] in all cells). Immediately you might notice that output is somewhat nonsensical. There are negative speeds in there! Those occur because of the physical impossibility of making up more time than the drive would take. You can't make up an hour on a 1 h drive, because you'd be going infinitely fast. You can't make up 2 h either (hence the negative speed).
One way to get this out of the way is to do an "if" gate, where a negative value returns a ridiculously fast speed, of, say 1000 MPH. This value will never appear on our graphs, but gives them the appearance of the infinite asymptote they are approaching. Copy cells B4:B12 and paste special "values" in B14. For cell C14 enter [=IF(C4<0,1000,C4)] and complete the column. Complete the row. Yay! Graph that!
Looks like at freeway speeds it is tough to make up time. Anything under 10 miles and you need some major speeding (the kind that can get you pulled over even in NJ) to make up just 2 minutes. Alternatively, drives over 100 miles lead to nice time gains from your speeding, such that you might even catch up on your TV watching.
Things get a bit more interesting on surface streets though. Bump the reference speed down to 30.
The lower speeds here make the curves look much sharper. Driving ten miles at our new reference speed would take about 20 minutes if we hit all green lights, but to shave even 5 minutes off that would require us to increase our speed by a third. On the other hand, that same level of speeding can save an hour when you go over a hundred miles.
Actually, now that I mention it, stoplights are a much better reason to be late. Forget that double-knotting of your shoe... hitting just one stoplight could put you a minute or two behind schedule. I wanted to see how much time you might lose to stoplights, so I played around with modeling traffic light timing in Excel before realizing:
A) It's hard.
B) It depends on what kind of pattern you have, and
C) It's been done before. Hundreds of thousands of times.
A much easier path is to make some assumptions:
Assumption 1. The number of lights you pass through increases linearly with distance you drive. (If you drive 10 miles you will pass through twice as many intersections as a 5 mile drive.)
Assumption 2. For every X lights you pass through, you will be stopped N number of minutes. (I'll choose 4 intersections, stopping for 1 minute. I don't know which intersections, just total stop time.)
Assumption 3. The spacing for intersections is similar to that in Seattle. Every 5 blocks = major intersection = quarter mile.
This results in something quite interesting. There is no effect of speed on the amount of time you are stopped at lights. Of course this is a little simplistic, and does not account for running yellow lights (which would favor speeding) or timed lights for a given stretch (which discourages speeding), but instead lets us calculate a new ratio of "stoplight time / mile". With the conditions I mentioned in assumption 2, that ratio would be 1 min/mile. Glancing back up at the 30 MPH chart, I can see that to save 1 min on 1 mile (or 10 min on 10 miles etc.) would be... 60 MPH?
Let me double check that. Ten miles at 30 MPH is 20 min. Adding 1 min/mile of stoplight time would bring the total to 30 min. If we tried to get back down to 20 min total trip time we'd subtract 10 min for sitting at stoplights and try to make the 10 miles in the remaining 10 minutes. Yup, thats 60 MPH.
And the sad thing is that I was trying to be conservative with my estimate of stoplight time. Downtown it could be much higher, and remember, this stoppage is distance independent, so we can't just drive 1000 mies to get efficiency (although after we leave the city the stoplight time/mile should drop significantly). And here's a spooky thought (my hedge against whether or not I get in a Halloween post), what if the stoplight time gets to 2 min/mile?
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