# Toboggan Speed Theory

1. Jan 26, 2008

### CoolClay

So I participate every year in the Toboggan National Championships in Maine. Year after year most teams try to get as fat as they can with the theory that the more weight they have the faster they will go down hill.

I am fairly proficient in basic physics, and know that the acceleration due to gravity always stays the same no matter what the mass, but friction increases with the added mass.

So to me the "more weight=better" theory is counterintuitive. From my understanding it would be more advantageous to have a team that is very light, and very aerodynamic, then a team that is big fat, and non-aerodynamic.

Is this right or am I missing something. Thanks for your help!

2. Jan 26, 2008

### Hootenanny

Staff Emeritus
Ignoring air resistance, the acceleration of a body down an incline is independent of mass. To illustrate this point consider a body of mass m, travelling down a slope inclined at an angle $\theta$ above the horizontal, then the weight of the sledge resolved down the slope is given by;

$$F_g = mg\sin\theta$$

And the frictional force [acting up the slope];

$$F_f = \mu R = -\mu mg\cos\theta$$

Hence, the net-force acting on the sledge parallel to the slope is;

$$F_{net} = mg\left(\sin\theta - \cos\theta\right)$$

Then, the acceleration of the sledge down the slope is given by;

$$a = \frac{F_{net}}{m} = g\left(\sin\theta - \cos\theta\right)$$

Which is independent of mass. However, if we take air resistance into account, then the acceleration has some mass dependence. Hope this helps.

Last edited: Jan 26, 2008
3. Jan 26, 2008

### greeniguana00

Also, the force due to air friction slowing down the tobogganers is dependent on speed and surface area, not mass, but the force due to the combination of the normal force of the slope and gravity will be greater for heavier people, meaning heavier people of the same size and shape would be affected less by air resistance.

4. Jan 26, 2008

### Hootenanny

Staff Emeritus
It is not the magnitude of the force that is important, but rather the acceleration.

5. Jan 26, 2008

### greeniguana00

Yes, and F=ma. Heavier people (assuming all else equal and that the fluid dynamics involved don't cause anything funky to happen) will accelerate faster than lighter people because the fixed force of air resistance will have less of an effect in terms of acceleration on the larger mass, but the force due to gravity and the normal force will cause the same acceleration for both masses.

6. Jan 28, 2008

### CoolClay

Ok so I think I understand all that now. So that would mean although the force due to gravity would remain the same for any mass, the normal force for a greater mass (F=ma) would be greater, which even with increased air resistance would still be greater acceleration then less mass.

So my theory is incorrect and in terms of tobogganing, the more mass, the more acceleration, so the commonly held belief is correct.

Correct?

7. Jan 28, 2008

### Hootenanny

Staff Emeritus
It is imperative here that you first understand the difference between force and acceleration. The weight or force due to gravity is dependent on mass, W = mg, but the acceleration due to gravity remains unchanged;

$$a = \frac{W}{m} = g$$

There is a difference and it is crucial that you understand it.

I said, right at the beginning of my post "ignoring air resistance". As both me and greeniguana00 have said, once air resistance is brought into the picture, the acceleration of a sledge becomes dependent on mass. In fact, the acceleration would be inversely proportional to the mass so a larger mass, would in fact increase your acceleration. However, as I am sure you know, air resistance is directly proportional to cross-sectional area, so streamlining then becomes important.

If I were to re-write the final equation in my last post to take into account air resistance it would be something of the form;

$$a = g\left(\sin\theta - \cos\theta\right) - \frac{kA}{m}v^2$$

Where k is a collection of constants, A is cross-sectional area, m is mass and v is velocity. So you see that a larger mass with the smallest cross-sectional area possible would be the optimum solution, but alas, heavier people tend to have a larger cross-sectional area in general.

I hope that clears things up.

Last edited: Jan 28, 2008
8. Jan 28, 2008

### Kittel Knight

Well, you meant acceleration instead force, didn't you ?

9. Jan 28, 2008

### NateTG

There is technology for reducing the friction of the bottom of the toboggan on the chute. That aspect of things is probably as, or more important that the air resistance.

An extreme approach would be to run a ground-effect toboggan with a bottle of compressed air, and there are almost certainly a litany of waxes, and you might get some sort of return out of dimpling sections of the bottom and modifying the shape of various parts.

10. Jan 7, 2012

### lezrati

I think I followed all this but don't we also need to consider the friction between the bottom of the toboggan and the snow. The pressure from the weight on the sled actually melts some of the snow. The sled slides on a thin sliver of water which creates less friction than the snow. The heavier guys create more melting, therefore more water and less friction

11. Jan 7, 2012

### tommw

lezrati is right. When it comes to sledging, friction makes all the difference.
Or why do you think it is forbidden to make your bob sledge heavier in olympic competitions although it wouldn't change air resistance when placed inside the bob sledge.

But apart from that you may also ask if heavier people roll down a hill faster on a bike than lighter people...

12. Jan 7, 2012

### mrspeedybob

You might also get some returns from designing your toboggan so that a highly heat conductive path exists between the tobogganers bodies and the ice. This would improve melting at the expense of comfort, but in a national championship I doubt comfort is a priority.