What is the terminal speed for an 77 kg skier going down a 45 ∘ snow-covered slope on wooden skis μk= 0.060?
Assume that the skier is 1.8 m tall and 0.40 m wide. Assume the skier's drag coefficient is 0.80.
Express your answer using two significant figures with the appropriate units.
To calculate the terminal speed of the skier, we need to consider the forces acting on them. The two main forces are gravity and drag.
First, let's find the gravitational force acting on the skier. The formula for gravitational force is given by:
F_gravity = m * g
Where:
m = mass of the skier = 77 kg
g = acceleration due to gravity = 9.8 m/s^2
F_gravity = 77 kg * 9.8 m/s^2
F_gravity = 754.6 N
Now, let's find the drag force acting on the skier. The formula for drag force is given by:
F_drag = (1/2) * ρ * v^2 * A * Cd
Where:
ρ = air density = 1.2 kg/m^3 (approximate value)
v = velocity of the skier
A = cross-sectional area of the skier = height * width = 1.8 m * 0.40 m = 0.72 m^2
Cd = drag coefficient = 0.80 (given)
Since we are looking for the terminal speed, the drag force will be equal to the gravitational force:
F_drag = F_gravity
Now we can solve for the terminal speed by rearranging the formula for drag force:
v^2 = (2 * F_gravity) / (ρ * A * Cd)
Substituting the given values:
v^2 = (2 * 754.6 N) / (1.2 kg/m^3 * 0.72 m^2 * 0.80)
v^2 = 1579.269 s^2/m^2
Taking the square root of both sides:
v = sqrt(1579.269 s^2/m^2)
v = 39.75 m/s
Therefore, the terminal speed for the skier going down the snow-covered slope is approximately 39.75 m/s.