Terminal velocity of a skier using a Momentum Balance

[Henrique Orlandini]

Homework Statement

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A skier (mass M = 100 kg) going down a slope with inclination θ = 30°, sliding in a fluid-like snow (viscosity μ = 100 mPa*s) of thickness h = 0.01 m, using a pair of skis, each one with a surface area of A_s = 0.15 m², reaches terminal velocity v_t after some distance. Air resistance (density ρ = 1.2 kg/m³) correspond to a drag coefficient (C_D = 0.2) which refers to the crouched skier area A_D.

a) Dermine the terminal velocity of the skier.
b) Dermine the terminal velocity of the skier without air resistance.
c) Dermine the terminal velocity of the skier without snow viscosity.

Additional info: g = 9.8 m/s².

Homework Equations

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Drag force: $F = C_D*ρ *(v^2/2)*A_D$
Weight: $F = M*g$
Momentum balance: $\dot {\mathbf L} = \sum \mathbf F_{ext} + \sum {\dot m} \mathbf v$

The Attempt at a Solution

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First I set the directions, with downward y and movement-forward x being positive. The I drew the FBD of the problem, with the weight of the skier going downward and drag force going against the movement, along the slope. Then I decomposed the drag force in x and y components, both negative according to my referential.

Using the momentum balance equation, I tried to solve the problem finding $v_x$ and $v_y$ so I could sum them to find $v_t$ in the slope direction.

For $x$ I found: $\dot {\mathbf L} = -C_D*ρ *({v_x}^2/2)*A_D*sinθ$, in which $\dot {\mathbf L} = 0$ equals zero because we're at terminal velocity, so there is no acceleration.

I found something similar to $y$, but with weight added: $\dot {\mathbf L} = -C_D*ρ *({v_x}^2/2)*A_D*cosθ + M*g$, with the same conclusion $\dot {\mathbf L} = 0$.

But only with those forces, I should get a zero speed in x and an absurdly high speed in y (about 255 m/s), which is clearly incorrect.

I also didn't use some parameters the problem gave me, such as h, μ and A_s, which all are related to "viscous tension", but from what I've read, drag force (and lift force) are the sum of pressure force and viscous tension, so putting tension in this FBD sounds wrong.

I think it's either this or $\dot {\mathbf L}$ isn't really zero, but I really cannot see a acceleration in a terminal velocity scenario.

 

[haruspex]

In your equations you have neglected normal force.