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Physics of snowboarding (2) Fri, Mar 20, 2009
Whenever I zoom past people on a snowboard, I wonder if my speed is a result of my recklessness, or if it's because of my weight.  The answer to this lies in the fun and exciting world of physics, which attempts to explain complex phenomena by assuming the world is ideal.  And away we go! 

If we treat the skier or snowboarder as an object on an inclined plane, we can simply sum the forces in each direction.  If we assume the direction of motion is x and the direction perpendicular to the hill is y, we have: 
Fx:  W*sin(theta) - Ff - Fd = m*ax
Fy:  Fn - W*cos(theta) = m*ay

W = weight = m*g
Ff = friction force = mu*Fn
Fd = drag force = 0.5*rho*cd*s*v^2
Fn = normal force

m = mass
g = acceleration due to gravity
theta = slope of hill
mu = coefficient of friction between snowboard and snow
rho = density of air
cd = drag coefficient of human
s = presented area of human
v = velocity
Since there's no motion perpendicular to the hill (unless you're jumping), the forces in the y-direction are zero.  Solving for normal force and doing some algebra, we get the acceleration in the x-direction as a function of several terms: 
a = g*sin(theta) - mu*g*cos(theta) - 0.5*rho*cd*s*v^2/m
These three components are gravity, friction, and drag.  If we take drag out of the equation for a moment (since it depends on velocity and doesn't contribute much for slow speeds), a snowboarder's motion down the hill is simply a function of gravity, slope, and coefficient of friction, which means it's not a function of mass, i.e. heavier people don't go faster. 

However, coefficient of friction is a tricky thing.  It depends on the amount of surface area in contact as well as the specific materials involved.  For example, if one ski contributes X amount of friction, two skis will contribute 2X.  And then there's the fact that skis and snowboards are rarely in full contact with the ground.  Also, board length and rider stance can each have an effect.  Finally, snowboard material, finish (including wax), and snow consistency all play a role.  Suffice it to say rider mass has an effect, whether or not it's significant or measurable. 

Getting back to drag, we can further assume that air density and drag coefficient are essentially constants (cd technically varies with velocity and body shape, but the values are all pretty close), which leaves just a few variable terms: 
a ~ s*v^2/m
Simplifying it even further, it can be shown that mass and area vary inversely and almost equally, i.e. a heavier person tends to have more surface area than a lighter person.  And since one is in the numerator and one is in the denominator, they essentially cancel out.  This leaves us with: 
a ~ v^2
In human terms, this says that a person's acceleration varies inversely with the square of their velocity, which in even more human terms means that the slower a person is going, the faster their acceleration will be, which doesn't necessarily say much, but it does say one thing:  Speed doesn't depend on mass.  So no, I don't go faster because I'm heavier.  I probably go faster because I'm a moderate adrenaline junkie and my enjoyment on a snowboard comes through speed, not carving back and forth down a mountain.  That being said, my weight has an effect on the coefficient of friction between my board and the snow, and while it's hard to quantify, can be assumed to have at least some affect on speed.  How's that for a non-answer? #science

Comments:
Rus Sat, Mar 21, 2009
Dave, you lost me just before the independent clause of your second sentence.  ;-)

Dave Sat, Mar 21, 2009
Since I don't know what an independent clause is, I'll assume that's your point.  Well played.


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