a 75.0 kilogram skier is at the top of a hill that is thirty degrees above the horizontal. the skier starts form and then travels down the hill which is 100 meters long. the skier at the bottom of the hill instantaneously then travels on flat horizontal snow. the coeefficient of sliding friction between the skier and the snow is 0.100. calculate the distance the skier stops from the bottom of the hill.
Assume the snow on the hill and horizontal have friction.
the Potential energy of the skier is 75g*100sin30
all of that energy is expended in the snow.
frictionworkon hill=mu*Fn*100
= mu*mgSin30*100
friction work on horizontal
= mu*mg*distance
set the sum of the friction work equal to the initial GPE, and solve for distance.
I had a brain freeze, change sin30 in the friction to cos30. Goodness
To calculate the distance the skier stops from the bottom of the hill, we need to consider the forces acting on the skier during the downhill descent and after reaching the flat snow.
1. Downhill Descent:
Since the hill's slope is given as 30 degrees above the horizontal, we need to resolve the forces acting on the skier into their components parallel and perpendicular to the slope.
- The gravitational force acting downward can be split into two components:
- The component parallel to the slope, known as the gravitational force down the slope.
- The component perpendicular to the slope, which essentially pushes the skier into the slope.
The downhill force component parallel to the slope is given by:
Force_parallel = m * g * sin(theta)
where:
m = mass of the skier = 75.0 kg
g = acceleration due to gravity = 9.8 m/s^2
theta = angle of the slope = 30 degrees
2. Friction force:
The friction force opposes the downhill motion of the skier. It can be calculated using the coefficient of sliding friction (μ) and the normal force acting on the skier.
The normal force is the component of the gravitational force perpendicular to the slope, given by:
Force_perpendicular = m * g * cos(theta)
The friction force can be calculated as:
Force_friction = μ * Force_perpendicular
3. Net force and acceleration:
Now, with the downhill force component parallel to the slope and the friction force, we can calculate the net force on the skier. Since the net force determines acceleration, we can use it to find the skier's acceleration down the hill.
Net Force = Force_parallel - Force_friction
Acceleration = Net Force / m
4. Distance traveled on the hill:
To find the distance traveled on the hill, we can use the following kinematic equation:
v^2 = u^2 + 2as
where:
v = final velocity of the skier on the hill (v = 0 because the skier stops at the bottom)
u = initial velocity of the skier on the hill (u = 0 because the skier starts from rest)
a = acceleration calculated from the previous step
s = distance traveled on the hill (unknown)
5. Distance traveled on the flat snow:
After reaching the bottom of the hill, the skier continues to slide on the flat snow for a certain distance until they stop. On the flat snow, the only force acting on the skier is the frictional force.
The frictional force on the flat snow is given by:
Force_friction_flat_snow = μ * (m * g)
To calculate the distance traveled on the flat snow, we can use the equation:
F_friction = μ * (m * g) = m * a
where:
F_friction = Force_friction_flat_snow
a = acceleration on the flat snow (unknown)
s_flat_snow = distance traveled on the flat snow (unknown)
6. Final step:
Finally, to find the total distance the skier stops from the bottom of the hill, we need to add the distance traveled on the hill (s) to the distance traveled on the flat snow (s_flat_snow).
Let's calculate all these values step by step.
Step 1: Calculate the downhill force component parallel to the slope:
Force_parallel = m * g * sin(theta)
Step 2: Calculate the friction force:
Force_perpendicular = m * g * cos(theta)
Force_friction = μ * Force_perpendicular
Step 3: Calculate the net force and acceleration:
Net Force = Force_parallel - Force_friction
Acceleration = Net Force / m
Step 4: Calculate the distance traveled on the hill:
v^2 = u^2 + 2as
Since the skier comes to rest (v = 0) and starts from rest (u = 0):
0 = 0 + 2as
s = 0.5 * (v^2 / a)
Step 5: Calculate the distance traveled on the flat snow:
Force_friction_flat_snow = μ * (m * g)
F_friction = m * a
Set the two equations equal since the skier comes to rest again:
μ * (m * g) = m * a
s_flat_snow = 0.5 * (v^2 / a)
Step 6: Calculate the total distance:
Total distance = s + s_flat_snow