A skier starts from rest at the top of a hill that is inclined at 11.0° with respect to the horizontal. The hillside is 190 m long, and the coefficient of friction between snow and skis is 0.0750. At the bottom of the hill, the snow is level and the coefficient of friction is unchanged. How far does the skier glide along the horizontal portion of the snow before coming to rest?
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To find the distance the skier glides along the horizontal portion of the snow before coming to rest, we need to analyze the forces acting on the skier.
Let's break down the problem step by step:
Step 1: Calculate the component of the weight acting parallel to the slope.
The component of the weight acting parallel to the slope is given by:
Force_parallel = m * g * sin(theta),
where m is the mass of the skier, g is the acceleration due to gravity (9.8 m/s^2), and theta is the angle of inclination (11.0°).
Step 2: Calculate the frictional force opposing the skier's motion.
The frictional force is given by:
Force_friction = coefficient_of_friction * Normal_force,
where the coefficient_of_friction is 0.0750 and Normal_force is the component of the weight acting perpendicular to the slope.
Step 3: Calculate the distance traveled along the slope.
The distance traveled along the slope is given by:
Distance_slope = length_of_slope * cos(theta),
where length_of_slope is 190 m and theta is the angle of inclination (11.0°).
Step 4: Calculate the work done against friction during the descent.
The work done against friction is given by:
Work_against_friction = Force_friction * Distance_slope.
Step 5: Calculate the distance traveled along the horizontal portion of the snow.
The horizontal distance traveled is equal to the work done against friction divided by the force parallel to the slope (assuming no other significant forces are acting):
Distance_horizontal = Work_against_friction / Force_parallel.
Now, let's plug in the values and calculate the distance traveled along the horizontal portion of the snow:
Force_parallel = m * g * sin(theta) = m * 9.8 * sin(11.0°).
Force_friction = coefficient_of_friction * Normal_force = 0.0750 * (m * g * cos(theta)).
Distance_slope = length_of_slope * cos(theta) = 190 * cos(11.0°).
Work_against_friction = Force_friction * Distance_slope = (0.0750 * (m * g * cos(theta))) * (190 * cos(11.0°)).
Distance_horizontal = Work_against_friction / Force_parallel.
After calculating these values, you can determine the distance traveled along the horizontal portion of the snow before coming to rest.