power to weight ratio and top speed

[Cold Fussion]

Let's start with the force free body diagram



From Newton's laws we know:


thus:


where:



is engine power.
is the tyre rolling resistance coefficient.

And thus our equation is:


Now we know that terminal velocity happens when and ;

Which leaves us with


Now the reason why the vehicle mass does not effectively contribute to vehicle top speed is because of a car tyre the road surface is very small (in the 0.005 range), so the magnitude of the drag force at high velocities is much greater than of the friction force (can be over 10x more), hence it effectively becomes zero.

Therefore we are left with:



Or to rephrase it:


And so vehicle terminal velocity does not depend on power to weight ratio.

[Erunanethiel]

Let's start with the force free body diagram

http://i.imgur.com/geNwWIT.png

From Newton's laws we know:


thus:


where:



is engine power.
is the tyre rolling resistance coefficient.

And thus our equation is:


Now we know that terminal velocity happens when and ;

Which leaves us with


Now the reason why the vehicle mass does not effectively contribute to vehicle top speed is because of a car tyre the road surface is very small (in the 0.005 range), so the magnitude of the drag force at high velocities is much greater than of the friction force (can be over 10x more), hence it effectively becomes zero.

Therefore we are left with:



Or to rephrase it:


And so vehicle terminal velocity does not depend on power to weight ratio.

Very good explanation, thank you very much.

And acceleration is dependant on power power to weight ratio still right? Not the outright power. So between 200-300kmh acceleration, where drag is high, the acceleration time of a car with 500bhp/500kg and the other car with 1000bhp/1000kg will be exactly the same, everything else being equal. Correct?

[rscsr]

Very good explanation, thank you very much.

And acceleration is dependant on power power to weight ratio still right? Not the outright power. So between 200-300kmh acceleration, where drag is high, the acceleration time of a car with 500bhp/500kg and the other car with 1000bhp/1000kg will be exactly the same, everything else being equal. Correct?

No. This is not correct. The 1000bhp/1000kg will always accelerate faster.

You can rearrange the equation for the acceleration to the following:



The first part on the right side is the same for both cars. But the second term on the right side is always smaller for the heavier car. Therefore the 1000bhp/1000kg car has a higher acceleration at every speed

[Brian Coat]

Here's a coffee time teaser for you.

If you increase the weight of the car (and change nothing else) ... the top speed could go up!

E.g. Load up a softly sprung, fast road car with (say) 4 very heavy people and the reduction in frontal area may offset the increase in tyre drag?

This brings a new dimension to the term "Fast Food".

[mzivtins]

Given how many variants of vehicles there are in the market, the only sweeping rule could be that it is purely dependent on the gearing of the vehicle, just like some have said.

The top speed at the RPM red-line of car/bike in its highest gear wont change, no matter how much weight-loss or power you throw at it.

F##king gearboxes, Oppressing our need for speed

[Erunanethiel]

Very good explanation, thank you very much.

And acceleration is dependant on power power to weight ratio still right? Not the outright power. So between 200-300kmh acceleration, where drag is high, the acceleration time of a car with 500bhp/500kg and the other car with 1000bhp/1000kg will be exactly the same, everything else being equal. Correct?

No. This is not correct. The 1000bhp/1000kg will always accelerate faster.

You can rearrange the equation for the acceleration to the following:



The first part on the right side is the same for both cars. But the second term on the right side is always smaller for the heavier car. Therefore the 1000bhp/1000kg car has a higher acceleration at every speed

There is a video of a race between a bugatti veyron grand sport and s1000rr from 100 kmh i think to maybe 250kmh and the bike takes it. It is on a dependable channel as well, so they probably are not modified. They are very close on power to weight ratio. But the car has 6 times the power. Something, somehow doesnt make sense

[rscsr]

Very good explanation, thank you very much.

And acceleration is dependant on power power to weight ratio still right? Not the outright power. So between 200-300kmh acceleration, where drag is high, the acceleration time of a car with 500bhp/500kg and the other car with 1000bhp/1000kg will be exactly the same, everything else being equal. Correct?

No. This is not correct. The 1000bhp/1000kg will always accelerate faster.

You can rearrange the equation for the acceleration to the following:



The first part on the right side is the same for both cars. But the second term on the right side is always smaller for the heavier car. Therefore the 1000bhp/1000kg car has a higher acceleration at every speed

There is a video of a race between a bugatti veyron grand sport and s1000rr from 100 kmh i think to maybe 250kmh and the bike takes it. It is on a dependable channel as well, so they probably are not modified. They are very close on power to weight ratio. But the car has 6 times the power. Something, somehow doesnt make sense

I assume you mean this video.


My first concern for this comparison is that the bike and the car don't have the same drag. I would assume that the has way less drag and therefore the Drag/mass term could end up to be the same for both vehicles.
We assumed constant power and even if the Bugatti and the bike have the same power, the characteristics of the power delivery are most likely not the same and the gearing is most likely set up for a completely different useage (400km/h+ for the Bugatti and most likely 360km/h or something like that for the bike)
And it seems to me, that they are pretty close over the complete run anyway.

[Erunanethiel]

No. This is not correct. The 1000bhp/1000kg will always accelerate faster.

You can rearrange the equation for the acceleration to the following:



The first part on the right side is the same for both cars. But the second term on the right side is always smaller for the heavier car. Therefore the 1000bhp/1000kg car has a higher acceleration at every speed

There is a video of a race between a bugatti veyron grand sport and s1000rr from 100 kmh i think to maybe 250kmh and the bike takes it. It is on a dependable channel as well, so they probably are not modified. They are very close on power to weight ratio. But the car has 6 times the power. Something, somehow doesnt make sense

I assume you mean this video.
https://www.youtube.com/watch?v=1Zj2h-3AeiM

My first concern for this comparison is that the bike and the car don't have the same drag. I would assume that the has way less drag and therefore the Drag/mass term could end up to be the same for both vehicles.
We assumed constant power and even if the Bugatti and the bike have the same power, the characteristics of the power delivery are most likely not the same and the gearing is most likely set up for a completely different useage (400km/h+ for the Bugatti and most likely 360km/h or something like that for the bike)
And it seems to me, that they are pretty close over the complete run anyway.

Well the gearing I dont think is that long, compared to the bike, since they do 0-100 at the same time.
I have found the CdA numbers for the two. For the bike it is 0.55 and for the Bugatti its 0.745. Is that enough to offset a 1000 horsepower disadvantage?

Thank you very much

[rscsr]

I made some quick calculations with the information on Wikipedia for those 2 vehicles.
The bike has a land speed record of 322km/h=89.4m/s with a power of 146kW. If we assume that this speed has been done at max Power we get a cdA=0.4 m². The bike has a weight of 204.5kg. I assumed it is driven by a driver of 80kg so we get a total mass of 284.5kg. So we get a total P/m=0.51kW/kg and the (0.5*cDA*rho)/m=0.0007029 m^-1 .

The Bugatti has 882kW and a top speed record of 431km/h=119.72m/s. It has a mass of 1888kg and in this case 2 passengers so about 2048kg. Therefore we get a of cDA=1.028m².
So our P/m=0.43 kW/kg and (0.5*cDA*rho)/m=0.000257 m^-1.

So the video shows about what the equations say. The bike has a higher initial acceleration and at a certain speed is passed by the Bugatti. (Especially as they stop the race at about 230km/h or so I think. And as the crossover point is at about 200km/h we don't see to much of that)

And the results plotted as acceleration over velocity:

velocity acceleration plot
velocity acceleration sheet

[Mikey_s]

This is a great thread and I tip my hat to the mathematicians...
As a biker I would add a just a couple of points to the debate;
At low speeds the car is likely to be traction limited (more power than grip) and acceleration is likely to be limited by tyres; Clearly in an F1 car the tyres will have better grip than a road car, but the outcome will be affected by the practical ability to apply the power. However, for practical purposes I think at the lower speeds neither bike, nor car would be able to use maximum power
At lower speeds the bike will be limited by the ability to keep the front wheel down against the torque reaction - this is seen in the early stage of the video when the biker pulls a wheelie under heavy acceleration. Moreover, in the case of my bike (ZZR1400) the traction control system will manage the power to avoid both wheelies and wheel spin... I'm not sure what gadgets the bike in the vid has to manage these factors, or the Bugatti.

All superbike manufacturers in Europe agreed to (electronically) limit the top speed to 300km/h, so for practical purposes the race stops there for most bikes. Apparently mine will do that in 5th gear, with a gear to spare, so I'm not sure what the terminal velocity might be.

I would definitely second the statements about gearing affecting the outcome; In a race to terminal velocity the gear ratios and numbers of gears will be an important factor, even with a quickshifter.

[mzivtins]

All superbike manufacturers in Europe agreed to (electronically) limit the top speed to 300km/h, so for practical purposes the race stops there for most bikes. Apparently mine will do that in 5th gear, with a gear to spare, so I'm not sure what the terminal velocity might be.

Yep, that 186mph rule!

A de-restricted 5 year old GSXR-1000 will out strip 200mph easily in 100% stock trim and 5 years of age.

On a de-restricted litre bike, going downhill in 6th will have you bouncing off the limiter (terminal speed)

But that adds gravity into the situation, this hasn't been mention above in any calculations

TBH i think 1bhp per 1kg should be a Laymen term for an almost unobtainable terminal speed in track situ

[Tommy Cookers]

Let's start with the force free body diagram

http://i.imgur.com/geNwWIT.png

From Newton's laws we know:


thus:

......

also the inertia of the rotating masses (ie wheels etc) is a resistive term with a substantial effect on acceleration
and the engine etc inertia of rotating masses is another significant resistive term, greater with lower gears
these resistive terms can be treated as equivalent mass(es) additional to the car's mass

[Cold Fussion]

That was a very simplistic model, if we were to incorporate rotating masses it would probably simpler to use energy methods to model it?

[Brian Coat]

The inertia effects in lowest gear will be significant I imagine. The calculation is not sophisticated.

Did I read on another thread that 12,000 rpm is only roughly 100 kph in first gear - or 5 mph/1000 rpm?
Anyone got a ballpark kg.m^2 for engine-speed inertia on these V6's? This inertia will dominate the others due to gearing.

[Greg Locock]

For a light road car in 1st gear the inertia effects represent about 30% of the total effective inertia. The car in question was this one, one of my first jobs ever was to do the performance and mpg estimates.

https://drivingtothefuture.wordpress.co ... land-ecv3/

The brain numbing tedium of working that lot out by hand encouraged me to write a performance simulation program, which i was glad to see was still in use several years after I had moved on.