# Does body weight affect the speed when going downhill on a mountain bike?

We know heavier objects fall faster when dropped at certain height. I was wondering if I am going downhill on my mountain bike without any peddling, will I travel faster or slower because I am fat?

• Have you heard of this experiment? en.wikipedia.org/wiki/… it might clarify your first statement. You should also look at Galileo's inclined plane experiments to answer your question. – innisfree Jul 13 '15 at 11:29
• @innisfree Air resistance is a significant part of cycling. – David Richerby Jul 13 '15 at 15:35
• To be clear, you're incorrect - heavier objects DO NOT fall faster than light objects, all other things being equal. – talrnu Jul 13 '15 at 16:45

Heavier objects do not fall faster per se. But for heavy objects the influence of the air resistance will be smaller, if they have a similar surface area compared to the light objects.

The answer depends on the properties of your tyres and the road. But on an even road the air resistance will typically dominate once you reach a certain speed (the friction of the wheels $F_W$ will be more or less independent of speed, but not of weight as a heavier person deforms the tyres more, generation more friction, but as it is not the dominant part we will ignore it for now).

The air resistance of a person will vary approximately like $m^{2/3}$ or $m^{1/3}$ in dependence of the mass. The air resistance in turbulent flow is given by $F_R = \frac 1 2 \rho c_D A v^2$, where $\rho$ is the density of the fluid, $c_D$ is the dimensionless drag coefficient depending on the form, $A$ is the area of the object perpendicular to the flow and $v$ the velocity relative to the fluid. Your mass scales like $L^3$, so your area scales like $L^2 = m^{2/3}$ assuming isotropic growth, the drag coefficent $c_D$ will be roughly independent of your weight but highly dependent on your position and clothing, which also influence your surface area).

Your acceleration will be given by: $$a = g \sin(\theta) + F_\text{W} - \frac 1 2 c_D \rho v^2 \frac{A}{m} = \text{const} - O(m^{-1/3}).$$

This means you are at an advantage if you are heavier (or rather: larger and therefore heavier), as the influence of the drag scales like $m^{-1/3}$.

If we assume that your weight is not distributed equally in all directions you gain even more. But still, as the range of typical human weights which a bike can support is from about $50\,\mathrm{kg} \ldots 150\,\mathrm{kg}$ a light person in a aerodynamic position with tight clothes will probably still be faster than a heavy person sitting upright (as they will reduce their area to a fraction and lower their $C_D$).

• Real mountain biking would probably see advantages for both small and large riders in different circumstances, particularly as regards momentum. Sometimes having more momentum will get you through rough patches more easily, sometimes having less momentum will allow you to brake and turn more quickly or sharply. This is really for proper mountan biking, however, on the terrain it was designed for rather than just on a flat road or commuting, etc. – J... Jul 13 '15 at 16:45
• @J... agreed. Another really important factor are energy losses due to an uneven road, which will depend in a very complicated fashion upon the weight of the rider: Do small pebbles skid or remain in place, how much energy will be dissipated in the tyres and the frame, how much in the rider (who is elastic and will therefore dissipate vibrations) ... – Sebastian Riese Jul 13 '15 at 20:55

No and yes. At first, your assumption is not quite correct. In vacuum, all masses fall at the same speed. The reason is the that the mass cancels in the equations of motion:

$ma=mg$

$a=\ddot{x}=g$

To be more precise: the inertial mass and the gravitational mass are the same. Therefore, they cancel.

However, things change when you take air resistance into account. Of course, a feather has much more air resistance than a stone, consequently the feather falls slower.

In the equations of motion, the air resistance is proportional to the square of the velocity and a geometric factor that describes the shape of your object/body only, but not to the mass.

More specific to your question: Although you have a higher mass, the air resistance of your body compared to the bodies of other persons may be quite similar. Especially if you try to make you as small as possible. This means that you will have an advantage compared to less heavier people, as long as you manage to have not much more air resistance than them.

## protected by Qmechanic♦Jul 13 '15 at 18:28

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