Tour de France physics question

[kevinc]

Inspired by the Tour De France I have been running a poll on my local cycling group website asking whether a 'heavy' bicycle rider would roll down a hill faster than a 'lighter' rider.

Assuming the two riders (one 50kg, one 100kg) are on identical bikes and are stopped on top of hill then roll down the hill with no pedaling and no brakes and no corners to negotiate who would arrive at the bottom of the hill first and what physics principle would be involved ?

 

My cycle mates swear 'black and blue' the heavy rider would get to the bottom first.

Thanks in advance,

kevinc

[Klaynos]

Do you want us to work through the maths?

 

If not, ignoring air resistance (the bigger bloke will have more), and friction (again the bigger person will have more), they will reach the bottom at the same moment.

 

If you drop two balls off of a crane, one made of Iron and one of light plastic they will hit the ground at the same time.

[kevinc]

Do you want us to work through the maths?

 

If not, ignoring air resistance (the bigger bloke will have more), and friction (again the bigger person will have more), they will reach the bottom at the same moment.

 

If you drop two balls off of a crane, one made of Iron and one of light plastic they will hit the ground at the same time.

 

Thanks Klaynos, no need to crunch the numbers, just looking for a general explanation. I agree with you but my cycle friends disagree completely.

[Klaynos]

Gravity applies a constant force /kg of 9.81m/s/s /kg so for every kg increase the force is increased so the acceleration stays the same.

[Sisyphus]

That's correct, but I'm not so sure you can completely disregard wind resistance in this case. The main factor, gravity, will have identical effect on both riders, but you want to know who wins. Correct me if I'm wrong, but I'm pretty sure that at the Tour de France they don't say, "Well, they were pretty close, so I guess it's a tie." (They're not physics students.)

 

So then, the question is who is more slowed by wind resistance. The bigger guy will experience more because he's pushing more air, but he'll also have twice as much force working against it (his weight), so my guess is he's slowed less, and he wins (but not by much).

[Klaynos]

It's impossible to say without the drag coefficients of both riders and both of their masses, as it'll be dependent on their ratios... so without knowing that you can't say which it'll be...

 

There's also a greater frictional force on the heavier rider...

[DJBruce]

Yes ignoring all wind resistance they would reach the finish at the exact same time. But if you take into account resistance I think the lighter bike might win.

[DrP]

 

So then, the question is who is more slowed by wind resistance. The bigger guy will experience more because he's pushing more air, but he'll also have twice as much force working against it (his weight), so my guess is he's slowed less, and he wins (but not by much).

 

That's what I would have expected. Yes their drag co-efficients will be different, but not as different as their force opposing it i.e. mg. I reckon the big guy would just win. (obviously I could be wrong as Klaynos suggests as I have not done any maths here and each individual case would be different) Now... uphill would be a completely different matter!!

[kevinc]

Thanks everyone for your input. The heavy guys are still adamant they will get to the bottom of the hill first.

Link to the cycle forum

http://groups.msn.com/hotgossipgreatexchanges/general.msnw?action=get_message&mview=0&ID_Message=7441&LastModified=4675681699094903619

[Mr Skeptic]

The big guy will have more drag, and an even tinier bit more friction, if the bike is any good, but will have a lot more potential energy, or momentum. So I think the heavier guy will win, much like a rock would beat a grain of sand when dropped, because though he will have more drag, it will slow him less.

[dirtyamerica]

How about 2 guys that are the same size (volume) but one has more mass. So one guy weighs more but isn't bigger and causing more air resistance. In that case, my guess is the guy with a higher density wins.

[insane_alien]

i think the heavier guy would win. volume goes up faster than surface area so the extra weight should help him ignore air resistance a little bit more than the light guy.

 

if they were both the exact same shape then it would be the heavier guy.

[hobz]

Perhaps we SHOULD go through the math, since there seems to be alot of "I think"'s?

[swansont]

You're going to have a force that looks something like [math]mg sin\theta - \sigma v^2[/math] so terminal velocity is [math]\sqrt{\frac{mg sin\theta}{\sigma}}[/math]

 

The heavier one also has more friction, though, and that hasn't been accounted for.

[dirtyamerica]

I ride bikes too. Actually, while in Colorado I got the chance to ride down a mountain on the highway with my mountain bike back in 2001. I brought with me a backpack full of heavy stuff assuming I would overcome a little more air resistance with the added weight tucked back there. I also hoped that the backpack being located behind me would negate any extra drag. I only rode down once so I have no experience doing the same ride sans weight.

 

Got up to ~50mph before I ran out of gears and reached terminal velocity. Gimme the right hill and I'd love to do 70.

[Sisyphus]

Perhaps we SHOULD go through the math, since there seems to be alot of "I think"'s?

 

Can't really do that without specific drag coefficients, friction, etc. You could just guess, but that would just be another "I think," and might not realistically reflect what two different bikers would actually experience. My guess would be that mass is roughly proportional to volume, while drag is roughly proportional to surface area. Weight, then, would increase faster than drag, favoring the bigger rider. I would also guess that other friction that would be negligible compared to air resistance in this scenario, but I don't know without actually going out and testing it.

[hobz]

Can't really do that without specific drag coefficients, friction, etc. You could just guess, but that would just be another "I think," and might not realistically reflect what two different bikers would actually experience. My guess would be that mass is roughly proportional to volume, while drag is roughly proportional to surface area. Weight, then, would increase faster than drag, favoring the bigger rider. I would also guess that other friction that would be negligible compared to air resistance in this scenario, but I don't know without actually going out and testing it.

 

You (not necessarily you, although your name suggest you are used to tedious work) could construct two functions of mass, drag coeff. and friction coeff. and compare the two. You could also assume equal (or negligible) drag- and friction coefficients, since the question was concerned with the mass.

 

Heavier riders usually have an advantage when going down hill compared to lighter riders. This advantage is almost the opposite of what the lighter riders have up hill. That is, the lighter riders reach the top first, but the heavier rider can catch up with him after reaching the foot of the hill again. This is based on my experience with watching the Tour on TV, and is not science.

[swansont]

Can't really do that without specific drag coefficients, friction, etc. You could just guess, but that would just be another "I think," and might not realistically reflect what two different bikers would actually experience. My guess would be that mass is roughly proportional to volume, while drag is roughly proportional to surface area. Weight, then, would increase faster than drag, favoring the bigger rider. I would also guess that other friction that would be negligible compared to air resistance in this scenario, but I don't know without actually going out and testing it.

 

Even this becomes more complicated because a person is not a sphere, and it's the cross-section presented to the wind that matters. So the mass relation to that particular cross section does not have a definite relationship: if we assume constant density, and cyclist A is more massive than cyclist B, it's going to depend how that extra mass manifests itself in height vs width vs depth.

 

And then, the cross-section is not the final say, because the details of the tuck will matter for the drag coefficient, and this will not be identical for differently-dimensioned riders.

 

Too many unknown details. Vague conditions beget a vague answer.

 

You (not necessarily you, although your name suggest you are used to tedious work) could construct two functions of mass, drag coeff. and friction coeff. and compare the two. You could also assume equal (or negligible) drag- and friction coefficients, since the question was concerned with the mass.

 

 

That's been addressed. If drag and friction are ignored, there is no mass dependence. Friction and drag account for all of the difference.

[John Cuthber]

I'ts a while since I rode a bike but I remember that even a slight headwind made a lot of difference to how easy it was to cycle. On the other hand, the bearings on a bike are pretty good so I guess that the effect of wind resistance dominates.

 

The fact that they are riders on bikes isn't very important if you are only looking at wind resistance. The problem is the same as two objects dropped through the air.

In that case, the small object will fall slower than the big one if the objects are otherwise similar.

As evidence I submit that fog doesn't fall out of the air quickly, but rain does. The small drops in fog are analagous to the smaller rider, the bigger raindrops act like the bigger rider.

[hobz]

That's been addressed. If drag and friction are ignored, there is no mass dependence. Friction and drag account for all of the difference.

The force calculation you did seems to dependent on mass?

[swansont]

The force calculation you did seems to dependent on mass?

 

That included drag.

[Big and Shiny]

Everyone's taking a very scientific approach to this whole conundrum which is great.

 

By way of introduction, I'm the hypothetical 100kg rider kevinc discusses, except it's closer to 120. As for "ignoring air resistance", we can't even get the local council to sweep the gravel off the shoulder let alone removing all the air and creating a laboratory quality vacuum. So we'll have to treat this as a real world problem and keep air resistance in.

 

The thing that seems to be missing from the discussion is the correct perspective. Everyone is very strong on the point that acceleration due to gravity is identical for both riders. I don't argue that fact. But it's not gravity that should be considered here, it's the gravitational force which is 100% mass dependent.

 

It is then necessary to call upon Newton's second law.

You're going to have a force that looks something like [math]mg sin\theta - \sigma v^2[/math] so terminal velocity is [math]\sqrt{\frac{mg sin\theta}{\sigma}}[/math]

 

In the absence of a value for [math]\sigma[/math] I will make a reasonable assumption; for a rider of any given height, their frontal area, and therefore drag coefficient, when properly tucked over the handlebars will be almost identical and also independent of mass, within a reasonable margin of error.

 

In the expression [math]\sqrt{\frac{mg sin\theta}{\sigma}}[/math] and with the above assumption in place, the terminal velocity is dependent on mass only.

 

Anyway, it all becomes irrelevant on August 10. After a club race there will be a downhill event. No drive chain on the bike, pure gravity. kevinc and myself are bringing together a group of riders and eliminating as many variables as possible. Friction will not be an issue as each rider will use the same bike. With the seat and handlebar height unchanged for each rider, shoulder and hip height/alignment and therefore drag should be fairly constant. Each rider will be given 3 timed passes (hopefully, time permitting) down a straight smooth 1km 5% descent.

 

I was going to drop a cheeky "if you choose to ignore quantum effects" into paragraph 3 until I remembered my audience and realised I'm probably closer to the centre of the IQ curve here than anywhere else I live.

[swansont]

The thing that seems to be missing from the discussion is the correct perspective. Everyone is very strong on the point that acceleration due to gravity is identical for both riders. I don't argue that fact. But it's not gravity that should be considered here, it's the gravitational force which is 100% mass dependent.

 

 

 

This is a common misconception.

 

F = ma, which means that the acceleration is F/m. More mass means more force, but it also means that the force has to accelerate more mass, and the two cancel out. (gravitational mass and inertial mass being identical) Klaynos pointed this out very early in the discussion.

 

The motion from the gravitational force is independent of mass.

[Bignose]

Just to back up what swansont said here, let's look at the math a little closer.

 

So, you have [math]F_g[/math] the gravity force. [math]F_g = mg[/math]. [math]F_g[/math] becomes the [math]F[/math] in [math]F = ma[/math], so [math]ma = mg[/math] or you can cancel the masses out and you get [math]a=g[/math]. Now, acceleration is the second derivative of position with respect to time so [math]a = \frac{d^2 x}{dt^2} = g[/math] and that's what you'll integrate to get how long it takes to get to the bottom. Note that the last equation there is completely independent of mass.

[J.C.MacSwell]

Assuming equally symmetric riders of the same density the heavy guy wins. The drag coefficient is probably the same or very close and the volume of the riders is proportional to their weight.

 

The frontal area of the heavier guy is, however, proportionally less.

 

Advantage heavy guy!