What is the terminal speed for an 77.0kg skier going down a 44.0 degrees snow-covered slope on wooden skis uk= 0.060?
Assume that the skier is 1.70m tall and 0.300m wide.
To calculate the terminal speed of the skier going down the slope, we need to consider the forces acting on the skier. The two main forces to consider are the gravitational force and the frictional force opposing the skier's motion.
First, let's calculate the gravitational force:
Gravitational Force (Fg) = mass (m) * acceleration due to gravity (g)
Given:
Mass of skier (m) = 77.0 kg
Acceleration due to gravity (g) = 9.8 m/s²
Fg = 77.0 kg * 9.8 m/s²
Fg = 754.6 N
Next, let's calculate the frictional force using the coefficient of kinetic friction (uk):
Frictional Force (Ff) = uk * Normal Force
Normal Force (N) = weight of the skier = m * g
N = 77.0 kg * 9.8 m/s²
N = 754.6 N
Ff = uk * 754.6 N
Ff = 0.060 * 754.6 N
Ff = 45.3 N
Since the skier is moving downhill, the frictional force will act in the opposite direction of motion, opposing the gravitational force. Therefore, we need to subtract the frictional force from the gravitational force.
Net Force (Fnet) = Fg - Ff
Fnet = 754.6 N - 45.3 N
Fnet = 709.3 N
Now, let's use the net force to calculate the terminal speed using the equation:
Fnet = Drag Force (Fd) = 0.5 * air density * cross-sectional area * drag coefficient * velocity²
Since we are solving for terminal velocity, the net force will be zero, and the drag force is equal to the gravitational force. Rearranging the equation:
0.5 * air density * cross-sectional area * drag coefficient * velocity² = Fnet = 709.3 N
To find the Air density, we need to know the conditions (e.g., altitude, temperature). For simplicity, let's assume the air density is 1.2 kg/m³.
Now, let's solve for velocity:
velocity² = (2 * Fnet) / (air density * cross-sectional area * drag coefficient)
Given:
Air density = 1.2 kg/m³
Cross-sectional area = height (1.70 m) * width (0.300 m) = 0.51 m²
Drag coefficient = uk = 0.060
velocity² = (2 * 709.3 N) / (1.2 kg/m³ * 0.51 m² * 0.060)
velocity² = 1182.16
velocity = √1182.16
velocity ≈ 34.4 m/s
Therefore, the terminal speed for the skier going down the slope is approximately 34.4 m/s.