Friday Puzzler (1/3/13)

serendip

Assume two cyclists (A and . A and B and their bikes are identical in all ways, except as stated below. In particular, the riders and their bikes are of the same mass.

The single difference is that A's wheels are lighter than B's and his bike's frame is correspondingly heavier. However, the masses of the bikes as a whole are identical.

The cyclists both begin at the top of the side of a valley (illustrated below) and roll down the side and up the other side. Each, when he comes to a stop, instantaneously turns around and rolls back down the valley and up the other side again, and so on.

Eventually, both riders come to a complete stop in the base of the valley.

Which rider stops first? And which travels farthest?

Lumen

The rider with heavier wheels goes further, because the wheels act as a temporary store of angular kinetic energy, so the rider accelerates more slowly, thus attaining a lower top speed on each cycle, thus losing less energy to aero drag on each cycle.

kenmc

the ballsiest one wins

serendip

kenmc wrote: »

the ballsiest one.

Funnily enough, their balls are identical.

dave_o_brien

They will arrive at the same time.

Diarmuid

Lumen wrote: »

The rider with heavier wheels goes further, because the wheels act as a temporary store of angular kinetic energy, so the rider accelerates more slowly, thus attaining a lower top speed on each cycle, thus losing less energy to aero drag on each cycle.

eh?

What are the forces acting on the bikes? They are "roleing" down the hill so
1. gravity
2. friction (rolling resistance)
3. aero drag

Gravity is the same, mass is the same, the bikes are identical apart from the weight of the wheels so I'll assume rolling resistance and aero drag is the same. I don't see how the answer could be anything but they travel the same distance

Lusk Doyle

The one using peds goes furthest. Rider A looks suspiciously like an alias in a recently released report!

Karma

rider A with lighter wheels.

Lumen

Diarmuid wrote: »

aero drag is the same

Only if speed is the same. Slower is more efficient.

AstraMonti

The one with ceramic bearings :pac:

Plastik

Does one have an ultegra chain?

niceonetom

Genuine head-scratcher. Well done OP.

There's no mention of air resistance in the question so I'm not sure if we're to imagine this happening in the pure vacuum where most of these question exist or in the real world (with instantaneous changes in direction aside)... and, truth be told I'm not immediately sure what effect drag would have anyway. Presumably the wheels, while weighing different amounts, are aerodynamically identical?

My first thought is that when the two riders cross the bottom of the valley for the first time the one with the heavier wheels would be carrying more angular momentum (a flywheel of sorts) and so would be likely to tralel further up the opposite side of the valley. So he would go further. At least at first. Maybe he'd come to rest sooner though for some reason I can't think of.

Now I have to think about time...

Okay, my initial guess is that they'd both take the same amount of time to come to rest but that heavy-wheels-man would, in that time, have travelled further.

Lumen

niceonetom wrote: »

My first thought is that when the two riders cross the bottom of the valley for the first time the one with the heavier wheels would be carrying more angular momentum (a flywheel of sorts) and so would be likely to tralel further up the opposite side of the valley.

No, because the heavier wheel takes more energy to spin up, the speed at the bottom of the hill will be lower. That is reversed when going uphill.

The heavy wheels act as a lossless damper. That's probably not the right word but Googling is prohibited.

jinkypolly

I reckon A travels furthest and possibly both stop at the same time.

ford2600

act as a temporary store of angular kinetic energy


Can you explain that one please? What information do you need to work out the "angular kinetic energy"?

Cape Clear

Well Done OP! Great little teaser for a Friday. Anyone ever try it out in apratical sense I wonder

Lumen

ford2600 wrote: »

Can you explain that one please?

Not without cheating.

ford2600

you'd need w (the greek version) and the mass.

Only problem being according to your post to last teaser w (the greek one) doesn't exist or at leasr is the same as v.....not sure exactly what you meant....

Lumen wrote: »

Not without cheating.

Lumen

ford2600 wrote: »

Only problem being according to your post to last teaser

I was wrong last time. This time I am aiming to be differently wrong.

colm_gti

Is the hill a strava segment, and do both rider A and B know that it is?

Lusk Doyle

colm_gti wrote: »

Is the hill a strava segment, and do both rider A and B know that it is?

Yes, and what sort of pussies are they that they only roll down and up?!? It's called cycling damn it, not rolling!

seamus

Actually thinking about it, the rider with the lighter wheels will stop first because he accelerates and decelerates more sharply than the heavier wheels.

I think the rider with the heavier wheels will travel further. But I'm not entirely sure why. My gut says that because they both start from the same position, the rider with the lighter wheels loses proportionally more distance each time they go up the other side than the heavier wheels do.

NeedMoreGears

Both riders start with the same potential energy. This gets converted into kinetic energy as gravity acts on the rider/bike. The kinetic energy is stored in two places (i) the overall speed of the rider/bike system relative to the ground and (ii) the additional rotational energy of the wheels. Think of a bike on a stand with the wheels are spinning - the overall system contains kinetic energy but it is not moving relative to the ground.

Lighter wheels, assuming the have the same centre of gravity as the heavy ones (which they should - unless one has a much heavier valve than the other) will contain less rotational kinetic energy for a given road speed - Given that both started with the smae potential energy, for the lighter wheeled bikek more energy will be contained in the overall forward velocity of the bike/rider system.

This should mean that the lighter wheeled combination will tend to try to get to a higher speed and consequently experience greater aero and rolling resistence. So the lighter wheeled bike gets to the bottom of the hill quicker but has less energy as an overall system than heavy wheeled one when it gets there. The reverse process occurs on the way back up the hill where the overall kinetic energy is used in climbing. Given that it had less energy at the bottom the lighter wheeled bike therefore cannot go as far back up the hill.

The process repeats with the lighter wheeled bike travelling less than the heavier wheeled one. Given that the lighter bike travels faster it comes to a stop first.


Where did I go wrong?

Konkers

NeedMoreGears wrote: »

Both riders start with the same potential energy. This gets converted into kinetic energy as gravity acts on the rider/bike. The kinetic energy is stored in two places (i) the overall speed of the rider/bike system relative to the ground and (ii) the additional rotational energy of the wheels. Think of a bike on a stand with the wheels are spinning - the overall system contains kinetic energy but it is not moving relative to the ground.

Lighter wheels, assuming the have the same centre of gravity as the heavy ones (which they should - unless one has a much heavier valve than the other) will contain less rotational kinetic energy for a given road speed - Given that both started with the smae potential energy, for the lighter wheeled bikek more energy will be contained in the overall forward velocity of the bike/rider system.

This should mean that the lighter wheeled combination will tend to try to get to a higher speed and consequently experience greater aero and rolling resistence. So the lighter wheeled bike gets to the bottom of the hill quicker but has less energy as an overall system than heavy wheeled one when it gets there. The reverse process occurs on the way back up the hill where the overall kinetic energy is used in climbing. Given that it had less energy at the bottom the lighter wheeled bike therefore cannot go as far back up the hill.

The process repeats with the lighter wheeled bike travelling less than the heavier wheeled one. Given that the lighter bike travels faster it comes to a stop first.


Where did I go wrong?

Does the extent of the effect depend on where the additional weight is. If its in the hub there will be less spinning force. If its in the rims, its will be greater. However the additional weight in the farme is evenly distributed.

So is weigth distribution is the key to the advantage that the rider with the heavier wheels may have?

Lumen

NeedMoreGears wrote: »

The reverse process occurs on the way back up the hill where the overall kinetic energy is used in climbing. Given that it had less energy at the bottom the lighter wheeled bike therefore cannot go as far back up the hill.

No. The difference in height between the start and end of each down-up cycle depends ONLY on the energy loss in that cycle, since at both end points there is zero kinetic energy.

mgh and all that.

NeedMoreGears

@lumen - think we're saying the same thing.

At the bottom of the hill the light wheeled bike has already lost more of its orginal mgh than the heavier one due to aero and rolling resistence forces. Hence it won't go as far up the far side. In addition because it's speed through the air is higher at the bottom of the hill it is losing energy faster than the heavy wheeler.

In one cycle (i.e down and back up) the lighter wheel version does lose more energy. It therefore travels less but at greater speed. Hence I think it stops first - whachyatink?

Lumen

NeedMoreGears wrote: »

whachyatink?

I share your conclusions, I'm just disputing the minutae. :pac:

lennymc

i dont know what a tracker mortgage is!

irishmotorist

I don't have a clue and base this on nothing, but I reckon they will both be the same. The heavier wheels, due to the 'flywheel' effect, will accelerate downhill faster but have a harder time going uphill - more negative momentum. The lighter wheels' forté will be on the uphill stretch.

The law of Swings and Roundabouts mean that they will cancel each other out.

Lumen

irishmotorist wrote: »

The heavier wheels, due to the 'flywheel' effect, will accelerate downhill faster

Unpossible. The flywheel is mopping up some of the gravitational P.E., so there's less going into acceleration. Or to put it another way, the flywheel is acting as a brake.

Inquitus

Is this taking place in NCD? if so they both run out of tubes after hitting too many potholes and the experiment doesn't complete.

jimmy the car

The cyclist who travelled furthest would of course be the one called Lance....

Zen0

Thanks OP, excellent puzzle. And thanks Need More Gears for rephrasing it in terms of potential energy converting to kinetic energy and back. I believe you are right. In a frictionless system both would stop in the same place, but because the one with the lighter wheels has a higher top speed, it will lose more energy to friction. I think!

velo.2010

Both riders/bikes act as Pendulums? Just seeing this now but I think niceonetom's explaination is fairly sound though maybe not recognising or saying they act as pendulums which might be what the OP is after perhaps?!

Just to be different I'll say they both travel the same distance and stop at the same time! edit: The weight of the wheels is throwing me but if both rider and bikes are the same mass, leave the same point and have to travel along the same path (amplitude?) then they travel the same length and stop at the same time. Most likely is that friction is having an effect due to the different wheel weights which slows the swing of the heavier wheel and it travels less but with the same stopping time as the lighter wheel. My Leaving Cert and first year physics are guiding me (wrongly no doubt).

Answer please!

kenmc

I think the the "travel" bit is a trick question.... they both start at the same point on the hill, and end up at the bottom. so the distance between start and end points are the same for both.
Q.E.D.

ford2600

One may cross the valley more often than the other?

kenmc wrote: »

I think the the "travel" bit is a trick question.... they both start at the same point on the hill, and end up at the bottom. so the distance between start and end points are the same for both.
Q.E.D.

kenmc

yeah but each time they go back towards their starting point, that's negative distance, so cancels out.

irishmotorist

Lumen wrote: »

Unpossible. The flywheel is mopping up some of the gravitational P.E., so there's less going into acceleration. Or to put it another way, the flywheel is acting as a brake.

I suppose that means that I'm not an accidental genius. Back to the day job so!

route66

kenmc wrote: »

I think the the "travel" bit is a trick question.... they both start at the same point on the hill, and end up at the bottom. so the distance between start and end points are the same for both.
Q.E.D.

Not Q.E.D. at all! The start and end points may be the same but one rider may travel further up the sides after the start point - and may do more iterations - than the other.

I guess that if the experiment was performed in a vacume the answer would be that both riders would be the same.

If not in a vacume: if the extra weight is all in the axel (not part of the extra rolling mass) of the bike with the heavier wheel, the answer would also be the same.

I guess as the extra weight moves towards the rim, then "Lumens Rule" above applies: the bike with the heavier wheels goes further.

I think.

kenmc

route66 wrote: »

Not Q.E.D. at all! The start and end points may be the same but one rider may travel further up the sides after the start point - and may do more iterations - than the other.

I guess that if the experiment was performed in a vacume the answer would be that both riders would be the same.

If not in a vacume: if the extra weight is all in the axel (not part of the extra rolling mass) of the bike with the heavier wheel, the answer would also be the same.

I guess as the extra weight moves towards the rim, then "Lumens Rule" above applies: the bike with the heavier wheels goes further.

I think.

But thats why it's a trick question. The question does not ask "which covers the most distance". It asks "which travels farthest". I maintain that they both "travel" the same distance in that they both start at point A and finish at point B, yet they may well cover different amount of distance in getting from A to B.

sconhome

Depends on who pumped their tyres before starting.

route66

kenmc wrote: »

But thats why it's a trick question. The question does not ask "which covers the most distance". It asks "which travels farthest". I maintain that they both "travel" the same distance in that they both start at point A and finish at point B, yet they may well cover different amount of distance in getting from A to B.

When I go out for a spin tomorrow, I hope to do about 40kms. I'll start at point "A" and finish at point "A". So I'll only travel 0 kms? Strava won't agree and I won't either ;-)

Maybe if the word "net" was used?

kenmc

route66 wrote: »

When I go out for a spin tomorrow, I hope to do about 40kms. I'll start at point "A" and finish at point "A". So I'll only travel 0 kms? Strava won't agree and I won't either ;-)

Maybe if the word "net" was used?

exactly. you will cycle 40km, you will cover a distance of 40km, but you will travel 0km.

1: to go from one place to another, as by car, train, plane, or ship; take a trip; journey: to travel for pleasure.
2: to move or go from one place or point to another.


It IS friday, after all.

serendip

OP here.

Lumen was on to the fundamentals of the problem in the very first response, and several other posters arrived at the correct answer.

Energy is always conserved.

In a no-friction, no-drag world, the only energies involved are potential energy, kinetic energy and rotational (or angular kinetic) energy. Think just about a single trip across the valley. At all heights, the bike with the heavier wheels has more of its energy locked up as rotational energy, so it has less kinetic energy. It goes slower, and therefore takes longer to cross the valley. Although both bikes do eventually reach the same height up the other side.

Now throw drag(*) into the picture. Drag is proportional to velocity squared. So a faster bike loses more energy to drag than a slower one. So, in the first trip across the valley, the faster, lighter-wheeled bike loses more energy to drag than the slower, heavier-wheeled one. The heavier-wheeled bike, therefore, reaches a higher point, and takes longer to do so.

On the second trip across the valley, the heavier-wheeled bike starts later, and starts with more potential energy, and so on. So the lighter-wheeled bike eventually stops first, and the heavier-wheeled bike eventually travels farthest.

Lumen: Perhaps you might work on a puzzler for next week?

Full disclosure ... I mentioned the Friday Puzzler from a couple of weeks ago to a friend, and it was he who came up with this problem. He's not a boardsie.

(*) I'm only considering drag here because the other sources of energy loss (friction, etc) are no worse than velocity squared, so it's enough to think just about drag.

route66

kenmc wrote: »

exactly. you will cycle 40km, you will cover a distance of 40km, but you will travel 0km.

1: to go from one place to another, as by car, train, plane, or ship; take a trip; journey: to travel for pleasure.
2: to move or go from one place or point to another.


It IS friday, after all.

Ok, but not all definitions state that you end up at a different place:
"to move in a given direction or path or through a given distance <the stylus travels in a groove>"

When people go on their travels - i.e. leave home to travel the world - most end up back at home when they are done. Based on the strict definition that is one interpretation of what traveling means, a round the world traveller wouls travel less than somebody who just walks to their front gate.

I don't agree and would like the world to accept my definition so that I am right and you are wrong.

It IS friday, after all.

kenmc

route66 wrote: »

Ok, but not all definitions state that you end up at a different place:
"to move in a given direction or path or through a given distance <the stylus travels in a groove>"

from the outside edge of the record to the inside. and then ends up back in the arm-rest

When people go on their travels - i.e. leave home to travel the world - most end up back at home when they are done. Based on the strict definition that is one interpretation of what traveling means, a round the world traveller wouls travel less than somebody who just walks to their front gate.

Only if the person who goes to their front gate stays there

route66

kenmc wrote: »

from the outside edge of the record to the inside. and then ends up back in the arm-rest

Only if the person who goes to their front gate stays there

I give up!

(I'm still right though ;-) )