# Motion/bike question

Okay here is the deal. I’m sitting on my trusty new mountain bike looking down a hill which at the bottom turns into an uphill climb exactly like the one I am getting ready to go down. Say each hill is 1500 feet in length and the grade of the hill is 30 degrees. How far would I get if I didn’t peddle my bike any and just let myself coast? Would I make it up the next hill or not? And if I don’t make it up the hill how far do I go and what is the top speed attained? For easiness’s sake let’s say the person weighs 180lbs. If I’m leaving out any other pertinent info I’m horribly sorry this has just been bugging me.

Some things you are leaving out:

A standard mountain bike has rather wide tires with deep tread. The friction that builds up on pavement at any decent cruising speed effectively prevents you from using a mountain bike for speed & distance riding. I do not believe you will get very far at all up that next hill.

I am used to a racing bike with very narrow & smooth tires, where I can get upwards of 40 mph downhill (I might be able to go faster, but my self preservation instincts kick in around 40-45 mph and I start to squeeze the brakes). One time I got on a mountain bike and nearly died trying to get any speed out of it. The grip on the pavement was so bad it felt like the brakes were rubbing.

Considering the influence of friction, I don’t know if it’s even possible to do the mechanical analysis you’re asking for.

Perhaps it would be easier to put it this way…

If there were no friction, and you didn’t pedal at all once you got going, then you would get up the 2nd hill exactly as high as you started from. That would happen regardless of your weight or the angle of slope. (This is analogous to a skateboarding tube, low friction there.)

Every bit of additional friction means you will get that much less up that second hill.

So your question becomes how to quantify the amount of friction, which is just about impossible to do. The path materials, the air density, the tires, moisture in the air, everything contributes.

Anything under your control that affects friction will affect the final result. Thus At’s points – thinner tires mean less friction, and so on.

IF you got to the top of the second hill, what you would have would be a perpeputal motion machine, which we know isn’t allowed. The factors that prevent you converting all your potential energy (at the top of the first hill) into kinetic energy (at the bottom of the hill) and back into an equivalent amount of potential energy (at the top of the second hill) are 1) friction of the tires on the ground, 2) wind resistance, and 3) friction in the wheel bearings. If you can determine values for these factors, you should be able to calculate how far up the hill you’ll get.

Wind resistance = Friction.

It’s all friction in different forms.

What’s so special about wheel bearings? As opposed to friciton from your bum bouncing on the seat, or foot contact with the peddles, or chains over the gears…

The speed you would attain at such a gradient would be sure to cause aerodynamic drag.

You would get partly up the hill but not all that far.

Beyond friction, there’s another “loss” many don’t tend to think about…this problem involves translation of the bike AND rotation of the tires.

Some of your potential energy (at the start) goes into overcoming the inertia (I) of the tires. In other words, the tires want to stay at rest and are opposed to rotation…regardless of friction. It takes a torque (T) to overcome the tires’ resistance to turning - just as any mass (m) requires a force (F) to overcome its resistance to being set into motion. Work must be performed to create the torque (T), and this energy is not converted back to potential energy at the end of the trip.

This is why a larger wheel is harder to start (or stop) turning as opposed to a smaller wheel - even if they were built to have the exact same mass (m), the inertia (I) is directly related to wheel diameter.

I’m a little rusty, but this is the rough story without sharpening our pencils…

• Jinx

Jinx, if there were no friction, the tires wouldn’t have to rotate, the bike could just slide down and up the hills. You could ride a transam on cement blocks up and down the hills, and it’d be the same. There’s no need for the tires to turn (OK, they’d have to rotate 60 degrees or so to keep the bike aligned properly).

I gotta go with: You won’t go up the second hill as high as you started, and you can’t mathematically quantify any further than that without going insane.