About Site



Socrates and Leadfoot
Written July 7, 2007


LEADFOOT   Pittsburgh drivers are stupid.

SOCRATES   That's a rather strong statement for so early in the day, my friend.  Why accuse your neighbors of stupidity?

L   They're bad drivers.  They don't know how to drive.  They even slow down to go through the tunnels!

S   Tunnels?  Are there many tunnels in Pittsburgh?

Squirrel Hill Tunnel (pghbridges.com)L   Several.  I'm talking mostly about the Fort Pitt and Squirrel Hill tunnels.  They're on expressways on the way into the city, where the road takes a shortcut underneath the hills rather than going around them. 

S   Does it not seem reasonable that when you drive your car beneath a mountain, you ought to reduce your speed?

L   There's no need to!  The road's just as good inside the tunnel as it is outside.

S   Then why do Pittsburghers slow down, do you suppose?

L   Beats me.  They must be claustrophobic or something.  They're afraid of the enclosed space.  Or maybe they're nervous about going into the dark.  

S   But you yourself don't get nervous?

L   Of course not.  I don't let up on the accelerator unless I have a good reason.

S   That's a prudent policy, isn't it?

L   And I don't complain about something unless I have a good reason.

S   Also commendable.  But you're complaining today.  Therefore, would I be correct in assuming that you must have been delayed at a tunnel recently?  Perhaps this morning?

L   I sure was.  The Squirrel Hill tunnel on the Parkway East.

S   So this morning you were driving on the Parkway — the above-ground portion — and then you arrived at the tunnel, correct?

L   That's right.

S   Before you reached the tunnel, how fast were you going?

L   About 60 miles an hour.

S   And how fast did you go through the tunnel?

L   Only about 40.

S   Hmm.  Let me understand this.  You were doing 60, and then you reached the tunnel and slowed down to 40?

L   Right.

S   Eureka!  I think we have found the problem!  Don't you see?

L   See what?

S   Are not you the bad driver?

L   Huh?

S   You slowed down at the tunnel!  Is that not precisely what you've accused the other drivers of doing?

L   But I had no choice!  If I hadn't taken my foot off the gas, I would have rammed into the back of the SUV ahead of me.

S   The vehicle ahead of you slowed down to 40?

L   Yes, so I had to do the same.

S   Eureka!  I think we have found the problem!  The bad driver is that guy in front  of you!  You didn't happen to get his license plate number, did you?

L   Well, no.  But he wasn't alone.  Everybody ahead of me was slowing down.

S   Everybody?  Whether they wanted to or not?

L   Right.

S   And why do you suppose they did that?

L   Because Pittsburgh drivers are stupid.

S   All of them?

L   Apparently.

S   Are you saying that all drivers except you are bad drivers?

L   They sure seem to be.  They could go 60 before; why couldn't they just keep going at the same speed through the tunnel?

S   Well, it does seem to be a mystery.  Let's see if we can solve it.  Tell me, was there a lot of traffic on the Parkway this morning?

L   There always is.

S   Hmm.  And does any of that traffic enter or leave the highway near the tunnel?

L   Sure.  There's an interchange just before it, and there's another one after it.

S   So some of the vehicles around you were trying to change lanes to reach an off-ramp?

L   Right.  And there were even more cars coming onto the Parkway there.

S   Were any of them merging into the lane you were using?

L   Sure.  They have to.  Once you get inside the tunnel, there's only two lanes in each direction.

S   Interesting.  Perhaps that gives us a clue.  Would you mind if I led you through a little thought experiment?

L   A what?

S   A couple of thought experiments, actually.  Are you with me?

L   I suppose so.

S   All right, here's Experiment One.  This one will be easy.  Let's imagine that you're driving down the Parkway at the head of a line of five cars.  They're all driven by members of your family.  Your wife is in the car behind you, then her brother, then her sister, and finally her cousin.  Okay?

L   Fine.  Yeah, keep my wife's cousin as far away from me as possible.  He's a real idiot.

S   Let's also imagine that traffic is light today, so your little group has spread out over a stretch of highway a mile long.  Your wife is a quarter mile behind you, her brother is a quarter mile behind her, and so on.  And all of you are going 60 miles an hour.  Got that?

L   Got it.  No traffic; must be a Sunday morning.

S   As I said, you're driving the lead car, and you're approaching the tunnel when a pickup truck pulls onto the highway in front of you.  Is he also doing 60 miles an hour?

L   Not right away, I suppose.  He has to accelerate up to speed.

S   Correct.  But you find yourself only a hundred feet behind him and closing fast.  Do you continue to drive 60?

L   Well, I'm not going to rear-end him.  I'll either swerve into the other lane or hit the brakes, I guess.

S   Let's say you slow down, but not very much.  By the time you get down to 57 miles per hour, you're no longer closing in on his tailgate, because he's picking up speed.  Both of you can now accelerate up to 60 again, correct?

L   Right.  It's just a temporary slowdown.

S   Now remember your wife, a quarter mile behind you.  She sees your brake lights come on.  Does she also slam on her brakes?

L   I doubt it.  She's got all that room in front of her, so she keeps on coming, but she never gets close to me.

S   Very good.  And her relatives behind her at quarter-mile intervals don't have to slow down either, do they?

L   Of course not.

S   So all of you sail through the tunnel at 60 miles per hour, correct?

L   Correct.  That's the way it should be.

S   But now let's consider the way it is.  May we move on to Experiment Two?

L   You're the professor.

S   This is a little more complicated.  This time, it isn't Sunday morning.  You and your wife and her relatives are not spread out over a mile.  You're much closer together, the way you would be in everyday traffic.  How close would you say your wife normally follows the car ahead of her?

L   Oh, I don't know.  She drives like anyone else.  Maybe fifty feet.

S   You're probably right, but at that speed, you're supposed to allow twice as much distance.  Did you know that?

L   Those rules come from the bureaucrats.  Or the insurance companies.  No one pays any attention to them, except some timid old geezers who shouldn't be out on the highway anyway.

S   If you say so.  Anyway, this time, when the pickup enters the Parkway in front of you and you slow down from 60 to 57, your wife is in her usual position fifty feet behind you.  Does she slow down too?

L   I guess she'd have to, or she'd run into me.

S   When does she slow down?  Right away?

L   Sure.  She doesn't have a lot of time to think about it.

S   But she does have to notice that you've hit your brakes, and she does have to decide that she ought to hit her brakes, and she does have to move her foot from the accelerator to the brake pedal.  Doesn't all this take a fraction of a second?

L   Well, yes, there's reaction time, I guess.

S   So by the time she actually starts to slow down, is she still fifty feet behind you?

L   She's closer than that.

S   Let's say twenty feet, okay?

L   Whatever.

S   And you've slowed down to 57.  Does she also slow down to 57?

L   Naturally.

S   But she's only twenty feet from your back bumper.  Won't she feel that's she's too close?

L   I suppose.

S   Won't she want to slow down a little more, to put a little more space between your car and hers?

L   All right, so she slows down a little more.  She gets down to 55 and lets me pull away from her a little bit.  But it's only for a moment, because I'm going to be getting back on the gas, and then she can too.

S   True.  But before that happens, let's consider her brother in the next car back.  He sees his sister hit her brakes and slow down to 55.  He comes up right behind her.  He has to hit his brakes as well.  He finds himself only twenty feet from her back bumper.  So he has to slow down not just to 55, but to 53, does he not?

L   I suppose.

S   And the fourth car in line has to slow to 51, and the fifth car to 49.  Do you see what's happening?

L   It appears that my little tap on the brakes is getting magnified.  I had to reduce my speed by three miles an hour, but my wife's idiot cousin in the fifth car has to slow down by eleven.

S   That's right.  And by the time he does slow down, you're already back up to speed, aren't you?

L   I'm doing 60 again, but my relatives are all bunched up.

S   So can a disturbance in the traffic flow last longer than the initial disruption?

L   I guess.  It's like when there's an accident.  The cops clear the wrecked cars away, but traffic is still backed up an hour later even though all the lanes are open.

S   Now think back to Experiment One.  You were the only driver who hit his brakes.  Everybody else maintained their speed.  But in Experiment Two, your relatives do have to slow down.  What has changed?

L   Our cars are closer to each other.

Squirrel Hill Tunnel (pghbridges.com)S   And why are they closer together?

L   Traffic is heavier.  There's not as much free space.

S   So when your wife's cousin, at the back of the line, slowed down to 49 miles per hour, was it because he was afraid of entering the tunnel?

L   Well, he is kind of a wimp, but I guess that's probably not the reason he slowed down.  It was because the rest of us were bunching up in front of him.

S   Therefore, what can we conclude is the true source of your complaint?

L   I used to blame other drivers for being too nervous to go into a tunnel at full speed.

S   And now?

L   I guess they're just too numerous to go in at full speed.

S   It's simple when you think about it, isn't it?



Back to Top
More Math/ScienceMore Math/Science