What is the terminal speed for an 81.0 kg skier going down a 41.0degree snow-covered slope on wooden skis mu_k= 0.060? Assume that the skier is 1.60 m tall and 0.500 m wide.
49.2
To calculate the terminal speed for the skier, we need to consider the forces acting on the skier while going down the slope. The two main forces are the gravitational force pulling the skier downhill and the frictional force opposing the motion.
First, let's calculate the gravitational force acting on the skier. The gravitational force formula is given by:
F_gravity = mass (m) * gravitational acceleration (g)
where m = 81.0 kg (mass of the skier) and g = 9.8 m/s² (acceleration due to gravity).
F_gravity = 81.0 kg * 9.8 m/s²
F_gravity = 794.8 N
Next, let's calculate the frictional force acting on the skier. The frictional force is given by:
F_friction = coefficient of friction (μ_k) * normal force (F_normal)
To find the normal force, we need to decompose the weight of the skier into components parallel and perpendicular to the slope.
F_parallel = F_gravity * sin(θ)
F_perpendicular = F_gravity * cos(θ)
where θ = 41.0 degrees (slope angle).
F_parallel = 794.8 N * sin(41.0°)
F_parallel = 510.4 N
F_perpendicular = 794.8 N * cos(41.0°)
F_perpendicular = 605.2 N
The normal force is equal to the perpendicular force, so:
F_normal = F_perpendicular
F_normal = 605.2 N
Now we can calculate the frictional force:
F_friction = μ_k * F_normal
F_friction = 0.060 * 605.2 N
F_friction = 36.3 N
The terminal speed is reached when the frictional force equals the gravitational force:
F_friction = F_gravity
Now let's calculate the terminal speed (v_terminal) using this equation:
F_friction = m * g = v_terminal * m^2
v_terminal = √(F_friction / (m^2))
v_terminal = √(36.3 N / (81.0 kg)^2)
v_terminal = √(36.3 N / 6561.0 kg²)
v_terminal ≈ 0.0606 m/s
Therefore, the terminal speed for the skier going down the slope is approximately 0.0606 m/s.