# Terminal velocity of a skier using a Momentum Balance

• Henrique Orlandini
In summary, the conversation discusses a skier going down a slope with various forces acting on them, including drag force, weight, and viscous tension. The goal is to determine the terminal velocity of the skier with and without air resistance and snow viscosity. A momentum balance equation is used to solve the problem, but there are discrepancies in the results. The problem also provides additional information such as the gravitational acceleration and the surface area of the skis. The issue may be due to neglecting the normal force or not accurately accounting for the viscous tension.
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 As = 0.15 m2, reaches terminal velocity vt after some distance. Air resistance (density ρ = 1.2 kg/m3) correspond to a drag coefficient (CD = 0.2) which refers to the crouched skier area AD.

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.

In your equations you have neglected normal force.

## 1. What is the definition of terminal velocity?

Terminal velocity is the highest constant speed that an object can reach when falling through a fluid, such as air or water. It occurs when the force of gravity pulling the object down is equal to the force of air resistance pushing against it.

## 2. How does the momentum balance equation relate to terminal velocity?

The momentum balance equation, also known as Newton's second law of motion, states that the net force acting on an object is equal to its mass multiplied by its acceleration. In the case of a skier reaching terminal velocity, the net force acting on the skier is zero, as the gravitational force and air resistance force cancel each other out.

## 3. What factors affect the terminal velocity of a skier?

The terminal velocity of a skier can be affected by several factors, including the skier's mass, the surface area of their body in contact with the air, and the density and viscosity of the air. Additionally, the shape and aerodynamics of the skier's equipment, such as their clothing and skis, can also impact their terminal velocity.

## 4. How is the terminal velocity of a skier calculated using the momentum balance equation?

The formula for calculating terminal velocity using the momentum balance equation is v = √(2mg/ρAC), where v is the terminal velocity, m is the mass of the skier, g is the acceleration due to gravity, ρ is the density of the air, A is the surface area of the skier, and C is the drag coefficient, which takes into account the shape and aerodynamics of the skier and their equipment.

## 5. Can a skier exceed their terminal velocity?

No, a skier cannot exceed their terminal velocity. Once the skier reaches their terminal velocity, the net force acting on them is zero, and they will continue to fall at a constant speed unless an external force, such as air turbulence, disrupts their motion.

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