Time Speed Distance question.

Let me try to articulate my question here.   Let's say the fastest way to cover one mile in a car is a straight line.  No we have to make a loop out of this straight line, is the fastest loop one that is two .5 mile straight lines with the shortest possible 180 at each end, or would it be one large circle?  Now scale that back to 1/2 or 1/4 mile.  

This seems highly dependent on the variable of the car, say using a shifter cart vs drag car vs sports car.  Is what I'm getting at here just the basis of skid pad testing?  Maybe I need to walk the dogs with someone to talk to.

When you say "fastest", I'm not good at geometry but two very sharp 180s at the end of a long straight doesn't sound fast at all.  I'm thinking circle.

In reply to Jerry :

Yeah, least time to cover the distance.

For a full mile loop clearly a circle would take the least time, but would that be true for a 1/4 mile loop.  Just thinking of a skidpad at any size would it be faster to make it ob round vs round.  Again it boils down to how bad a car can turn vs accelerate, seems pointless.  In the end it is just how one would enter and exit a corner followed by a straigt in a momentum car vs a very high powered car.

If I used my mustang and my Armada and started with a square track and gradually increased the corner radii both vehicles would hit a sweet spot of straight vs corner for the fastest lap.  A full mile I would guess that a circle will be same for both but as the square track gets smaller will there be a sweet spot where less than a circle is faster.

I'll stop now, and got test in the parking lot.

I used to use something like this as an example when teaching Evo schools.

Tight line vs carrying speed.

Imagine a square course, with dimensions anywhere from 10' on a side to 1000'.

The answer is not universal, depending on car and dimensions.  In particular, the re-acceleration rate of the car is key -- as is the corner radius.

This is the reason some autocross courses will favor one car over another.  Just saying

You'd need something with terrible corning and a ton of power to make it something other than a circle I'd think. For a 1 mile track a typical non aero car at 1.2g could do 123mph or about 29.3s. OTOH The world record for a 1/2 mile pass is around 13s. So you have about 3 seconds to turn around the car to beat a pretty average track day car.

The sharper the turn, the more speed you would have to scrub. The banking on the turns would also be a factor. 

Edit: I found an error in my math, which I probably should have double checked *before* I posted.....

A one mile perfect circle has a turn radius of 840'. Some quick math says that a 1 g steady state turn (roughly what a good car could pull) at that turn radius is ~110 mph.

At the other extreme would be two half mile straights with a really tight hairpin at each end.

Can a car average more than 110 mph down each half straight? A car would have to trap quite a bit more than 110 to make it average more than 110 down each straight.

Any other track configuration between the two is just going to be a sliding scale between those two extremes. So what I originally posted was wrong.  It's really a perfect circle that's going to be fastest for all but the fastest accelerating cars.

BA5 said:

A one mile perfect circle has a turn radius of 840'. Some quick math says that a 1 g steady state turn (roughly what a good car could pull) at that turn radius is ~65 mph.

 You have something wrong with your formula. Mapped to known radius corners at my local tracks that makes no sense. A 840' corner is BIG. That's in the neighborhood of the kink at Road America. (770' centerline to 1199' max) 65 is parade speed through there.

theruleslawyer said:

BA5 said:

A one mile perfect circle has a turn radius of 840'. Some quick math says that a 1 g steady state turn (roughly what a good car could pull) at that turn radius is ~65 mph.

 You have something wrong with your formula. Mapped to known radius corners at my local tracks that makes no sense. A 840' corner is BIG. That's in the neighborhood of the kink at Road America. (770' centerline to 1199' max) 65 is parade speed through there.

Yeah, I caught that after I posted.  The number was tickling my head a little funny: should probably check *before* I post. :D

It would seem that we're all in agreement that at the large extreme, the perfect circle is fastest and the only real question is if this continues to hold up through the small extreme. So let's take a look at that.

Anybody who has been driving for any period of time has hopefully (even if subconsciously) noticed that even at slow speeds there is tire scrub when turning at full lock. The most obvious manifestation of this is that the cars turning circle will continue to decrease the slower the car is going, all the way down to barely moving.

So let's start with a car whose absolute minimum turning circle at barely a crawl is 30', which is a distance of 94.25'. Now let's add 10' to that for 104.25'. So you can do two 30' half circles at say 1mph plus two 5' straights, or one 33.18' circle that can be run still at full lock but at perhaps 5mph... You can go try it in a parking lot and see, but I contend that the answer is still going to be a circle.

Driven5 said:

So we're all in agreement that at the large extreme, the perfect circle is fastest and the only real question is if this continues to hold up through the small extreme. So let's take a look at that.

Anybody who has been driving for any period of time has hopefully noticed that even at slow speeds there is tire scrub when turning at full lock. The most obvious manifestation of this is that the cars turning circle will continue to decrease the slower the car is going, all the way down to barely moving.

So let's start with a car whose absolute minimum turning circle (at barely a crawl) is 30', which is a distance of 94.25'. Now let's add 10' to that for 104.25'. So you can do two 30' half circles at 1mph plus two 5' straights, or one 33.18' circle that can be run still at full lock but at a significantly higher speed... You can go try it in a parking lot and see, but I contend that the answer is still going to be a circle.

I think the cornering math doesn't work real well in this condition. 1g-

30' circle- 15' radius = 15mph

33.18'= 15.77mph

I think you might need to open up the gap more to make sense. The rear wheels are basically doing nothing. In fact you'd probably want to pull the parking brake to turn. Not to slide, but to pivot. It would be interesting to go out and see how much real world minimum turn circle varies with a speed.

Circle.

As always, XKCD has covered it.

https://what-if.xkcd.com/116/

In reply to theruleslawyer :

Forget the 1g. No car will pull anywhere near 1g at the same time as consistently and continuously achieving it's minimum geometric turning circle, because the higher the cornering force ther more lateral slip the tires will necessarily have. The closer to 0g, the tighter the turn will be for a given steering angle. It also will not achieve enough acceleration on a 5' straight to make the parking brake a viable 'pivot' option, especially not consistently.

The whole point of taking it to an opposing extreme is to change the limiting variable(s). For a turn large enough to achieve max lateral force, the answer is circle. This is for a turn small enough that max lateral force cannot be achieved, the answer is still circle.

From a mathematical reference, the fastest path will have the LEAST VARIATION in acceleration. That will always be a circular path. The optimal track surface would be a funnel sort of shape that has increased banking as the radius is decreased. You may recall the change donation toys that were shaped like this and would keep a coin on edge on its entire path down the funnel.

Longitudinal acceleration would stabilize near zero as you could apply all the power all the time and effectively hold constant speed.

Andy Hollis said:

I used to use something like this as an example when teaching Evo schools.

Tight line vs carrying speed.

Imagine a square course, with dimensions anywhere from 10' on a side to 1000'.

The answer is not universal, depending on car and dimensions.  In particular, the re-acceleration rate of the car is key -- as is the corner radius.

That's the answer to a different question though. Changing the line also changes the distance. The racing line is simply trading distance for average speed to minimize time.

Assume the track is effectively only 1 car width. What shape of track achieves the highest average speed?

Driven5 said:

That's the answer to a different question though. Changing the line also changes the distance. The whole point of the racing line is trading distance for average speed to minimize time.

Assume the track is only 1 car width. What shape of track achieves the highest average speed?

It still depends on the car.  Imagine a Top Fuel drag racing car -- the fastest track for it is going to be something like Avus (two very long parallel straights right next to each other connected by very tight 180 degree corners), only even more extreme.

In reply to Driven5 :

Im not talking drifting. In a pivot the rear tire would provide resistance to moving forward and allow the car to use just the grip tangent to the turning circle. Take it to an extreme and imagine the inside rear wheel as fixed. The car would pivot about it so long a any component of the turning vector is sideways. Its more of an off road concept though. Ford even offers a special mode on the bronco to do this. https://www.ford.com/support/how-tos/more-vehicle-topics/steering-and-suspension/what-is-trail-turn-assist/

In reply to codrus (Forum Supporter) :

Yes, it is likely possible to create extremes enough to be exceptions to this, and top fuel may be one... But may not.

Don't forget to factor in the time and distance would take for them to slow to a near stop and turn around twice too. It's not outside the realm of possibility to me that even a Top Fuel car might be better off on a 1 mile circle than going down and back on a 1/2 mile drag strip with only 1000' of acceleration each way... Even neglecting everything else that needs to be done with the car before it's ready to run a second time, preventing it from performing the second half of the loop in similar fashion.

Driven5 said:

Yes, it is likely possible to create extremes enough to be exceptions to this, and top fuel may be one... But may not.

You can learn a lot about a problem by looking at the extremes -- it is a quick way to learn about the factors that affect the outcome.  Generally speaking if the extreme case is different then there will be non-extreme cases where those same factors matter but in a less-obvious way.

In reply to codrus (Forum Supporter) :

Agreed. That's why the main takeaway I see from all of this is: The harder a car can accelerate relative to how hard it can corner, the less it is disadvantaged over a given distance by taking a non-circular path.

A one mile "wall of death" track with a car having huge horsepower.