Time Speed Distance question.

This is like the reverse of lap-time simulation (e.g. OptimumLap or the serious $$$$ pro-level software): rather than "what is the fastest a car should be able to go on a given track?", we're trying to ask "what is the fastest track for a given car?".

Or something.  The fundamentals are similar anyway.  (I mean, at the extreme of the fundamentals, it's Newtonian physics...)

A while ago I dug up a bunch of info on lap time sim, and a few open-source or home-brew FSAE projects to copy, but I still have not made myself sit down and do anything with it.

A very old GRM thread

https://grassrootsmotorsports.com/forum/grm/free-track-simulation-software/16003/page2/

This thread exploded, didn't it.

And here my assumption wasn't "what's the fastest shape to travel one mile and return to the starting point" but rather "what's the fastest shape to go to a point a half mile away and back".

I imagine that the answer to the correct question will be a circle or something tri-lobed depending on how hard the car can accelerate at higher speeds vs how hard it can corner.  The corners should be shaped so that corner entry uses cornering forces to slow the car, not braking... which would seem to imply a circle.

Another thing that implies a circle is that slowing down requires much more acceleration to make up for the lost speed, let alone surpassing it.  Think of the riddle about driving a mile at 15mph, how fast would you have to drive a second mile to average 30mph over those two miles?  (The answer is that it's impossible)

SV reX said:

It's a circle. And I can't think of any kind of exception. 
 

A top fuel car is gonna be the fastest in a straight line, but it's gonna totally suck trying to get it turned around. You'll have to start slowing it down long before you ever get to top speed.

The compromise for the top fuel (or any car) is the speed at which it can maintain the highest consistent speed before loosing lateral grip. That's a circle- which some cars will take faster than others.

I wanted to revisit this because I was also in the circle camp.

I decided to make a simple model using this approach - the track sections either perfectly straight or a perfect half-circle. This gives two straights and two half-circles. I give it a mass, horsepower, drag-limited top speed (this determines the drag coefficient, too), and grip. Grip is broken out into braking, lateral, and longitudinal. 

Here are the results for some 3,000 pound car with a theoretical top speed of 225 mph if it had 1,000 hp. Braking and longitudinal acceleration are limited to 1.0 G and are constant. As the car accelerates down the straights (starting from the max speed of the corner or the car's top speed, whichever is lower), it checks the braking distance and starts slowing when it needs to reach the corner entry speed. Corner entry, exit, and apex are constat speeds. 

Pink is the only car that prefers two straights and is pretty much the dragster of the group with an abysmal lateral G limit of 0.2G but 1,000hp. The rest (red ever so slightly prefers a circle) are in the circle camp. This does show that there are extreme cases where the circle does not win, and those cases don't have to be as extreme as a top fueler, but I don't think even a T2K car would corner slowly enough for the straight track to be favorable over a circle. 

In reply to cyow5 :

0.2g is TERRIBLE too. For reference the federal highway administration lists the limit for a fully laden semi as 0.38g Most passenger cars etc are closer to that 0.7-0.9 range.

In reply to theruleslawyer :

I know - the point was to bound the problem. A top fueller probably is somewhere in the 0.2 ballpark though if it were to turn as tightly as possible. It shows that, technically, a circle is not always the answer, but it effectively is. 

In reply to cyow5 :

Don't forget to allow time for the guy in the top fuel dragster to get out and repack his chute after turn 1! 😝😝

Can he drive a full mile without an engine rebuild? (including 2 separate decel/ accel cycles)

cyow5 said:

In reply to theruleslawyer :

I know - the point was to bound the problem. A top fueller probably is somewhere in the 0.2 ballpark though if it were to turn as tightly as possible. It shows that, technically, a circle is not always the answer, but it effectively is. 

Oh sure. I'm just saying even for a top fuel dragster it's probably unrealistically low given known lower bounds. Realistically the quickest way to turn something like that is probably light up the tires at full lock. That doesn't really fit neatly in a simple formula though.

In reply to theruleslawyer :

Haha, nope. It's up there with having to repack the chute. 

I am not sure what this exercise is getting at.   They have high banked tracks that are run flat out -

A circular track that is 5280 feet in length has a radius of about 840 feet  (circumference = 2 (pi) r  or (pi) D. 

 So a circle 1680 feet in diameter.   

Put something like an Indy car on it, with tires that can handle the g load, and you could go pretty fast.  Probably so fast the driver would pass out from the g load unless he was wearing a G-Suit.

Also see The Slinger Speedway but its only 1/4 mile.

In reply to jharry3 :

No one doubts banking is faster; the question is if a circle is always going to be the fastest closed shape to cover a one mile distance. The answer is yes, outside of some comically extreme examples, so it is interesting there could be an exception.