An Olympic skier moving at 20.0 m/s down a 30.0 degree slope encounters a region of wet snow and slides 145 m before coming to a halt. What is the coefficient of friction between the skis and the snow?
0.74
To find the coefficient of friction between the skis and the snow, we can use the following steps:
Step 1: Find the acceleration of the skier.
Using the given values, we can find the acceleration of the skier using the equation:
acceleration = (final velocity^2 - initial velocity^2) / (2 * distance)
Here, the final velocity is 0 m/s (since the skier comes to a halt) and the initial velocity is 20.0 m/s.
acceleration = (0^2 - 20.0^2) / (2 * 145)
acceleration = -400 / 290
acceleration ≈ -1.379 m/s^2 (negative sign indicates deceleration)
Step 2: Find the gravitational force acting on the skier.
The gravitational force acting on the skier can be calculated using the formula:
force_gravity = mass * gravity
where mass is the mass of the skier and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 3: Find the normal force.
The normal force is the force exerted by the surface perpendicular to the skis. On a slope, it can be calculated using:
normal force = force_gravity * cos(angle)
where angle is the angle of the slope (30 degrees).
Step 4: Find the frictional force.
The frictional force can be calculated using:
frictional force = coefficient of friction * normal force
Step 5: Determine the coefficient of friction.
Finally, we can find the coefficient of friction by rearranging the equation from step 4:
coefficient of friction = frictional force / normal force
Now we can plug in the values to find the coefficient of friction.
To find the coefficient of friction between the skis and the snow, we can start by calculating the acceleration of the skier on the slope using the given information.
We know that the initial velocity of the skier is 20.0 m/s, and the skier comes to a halt after sliding a distance of 145 m. We can use the following equation to find the acceleration:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, since the skier comes to a halt)
u = initial velocity (20.0 m/s)
a = acceleration
s = displacement (145 m)
Plugging in the known values and solving for acceleration:
0^2 = (20.0)^2 + 2a(145)
0 = 400 + 290a
Simplifying the equation:
290a = -400
a ≈ -1.379 m/s^2
The negative sign indicates that the acceleration is in the opposite direction to the skier's motion.
Now, we can use the equation for acceleration on an inclined plane to find the component of the gravitational force parallel to the slope:
a = gsinθ - μgcosθ
Where:
g = acceleration due to gravity (9.8 m/s^2)
θ = angle of the slope (30 degrees)
μ = coefficient of friction (what we want to find)
Plugging in the known values and solving for the coefficient of friction:
-1.379 = (9.8)sin(30) - μ(9.8)cos(30)
-1.379 = (9.8)(0.5) - μ(9.8)(√3/2)
-1.379 = 4.9 - μ(9.8)(√3/2)
-6.279 = -4.9μ(√3/2)
μ(√3/2) = (-6.279)/(-4.9)
μ ≈ 0.785
Therefore, the coefficient of friction between the skis and the snow is approximately 0.785.
Initial K.E. = work done against friction while slowing down
= M g cos 30*mu
= (M/2)*V^2
M cancels out. Solve for mu