A skier traveling 12.0 m/s reaches the foot of a steady upward 18 degree incline and glides 12.2 m up along this slope before coming to rest. What was the average coefficient of friction?
The process is correct.
To find the average coefficient of friction, we need to use the concept of work-energy. The work-energy principle states that the work done on an object is equal to its change in kinetic energy.
In this case, the skier starts with kinetic energy when they reach the foot of the incline and then comes to rest at the end of the glide. Therefore, the work done by friction can be equated to the change in kinetic energy.
1. Find the initial kinetic energy:
The initial kinetic energy is given by the formula:
K1 = (1/2)mv^2
where m is the mass of the skier and v is the initial velocity.
No information is provided about the mass of the skier. Therefore, we cannot determine the actual value of the initial kinetic energy.
2. Find the final kinetic energy:
The final kinetic energy is given by the formula:
K2 = (1/2)mv^2
where m is the mass of the skier and v is the final velocity, which is zero in this case since the skier comes to rest.
Since the skier comes to rest, the final kinetic energy (K2) is equal to zero.
3. Determine the work done by friction:
The work done by friction is given by the formula:
Work = force * distance * cos(theta)
where force is the friction force, distance is the distance traveled, and theta is the angle of the incline.
In this case, the work done by friction is equal to the change in kinetic energy:
Work = K2 - K1 = 0 - K1 = -K1
Since the cos(theta) takes into account the incline angle, we can rewrite the equation as:
Work = -K1 * cos(theta)
4. Calculate the displacement along the incline:
The displacement along the incline is given by the distance traveled multiplied by the sine of the angle of the incline:
displacement = distance * sin(theta)
In this case, the displacement along the incline is given by:
displacement = 12.2 m * sin(18 degrees)
5. Determine the work done by friction in terms of the displacement along the incline:
Work = force * displacement
which can be rewritten as:
Work = force * distance * sin(theta) = -K1 * cos(theta)
6. Calculate the friction force:
force = (K1 * cos(theta)) / (distance * sin(theta))
Since the force due to friction is equal to the coefficient of friction multiplied by the normal force, we can rewrite it as:
force = (μ * m * g * cos(theta)) / (distance * sin(theta))
where μ is the coefficient of friction and g is the acceleration due to gravity.
7. Determine the average coefficient of friction:
To find the average coefficient of friction, we need to divide the friction force by the weight (m * g):
μ average = force / (m * g) = ((μ * m * g * cos(theta)) / (distance * sin(theta))) / (m * g) = (μ * cos(theta)) / (distance * sin(theta))
Note that the mass of the skier cancels out.
8. Substitute the known values and calculate the average coefficient of friction:
μ average = (μ * cos(18 degrees)) / (12.2 m * sin(18 degrees))
For the actual value of the average coefficient of friction, we need the value of μ. Without any further information, we cannot determine the actual value.
To find the average coefficient of friction, we can use the concept of work and energy.
Let's break down the problem step by step:
Step 1: Find the initial kinetic energy of the skier.
The initial kinetic energy (KE_initial) of the skier is given by the formula:
KE_initial = (1/2) * m * v^2
where m is the mass of the skier and v is the initial velocity.
Given information:
Initial velocity (v) = 12.0 m/s
Step 2: Find the final potential energy of the skier.
At the top of the incline, the skier comes to rest. Hence, all the initial kinetic energy is converted into potential energy at the top.
The final potential energy (PE_final) of the skier is given by the formula:
PE_final = m * g * h
where m is the mass of the skier, g is the acceleration due to gravity, and h is the vertical distance the skier traveled along the slope.
Given information:
Vertical distance (h) = 12.2 m
Angle of the incline (θ) = 18 degrees
Step 3: Find the gravitational potential energy at the top.
The gravitational potential energy (PE_gravity) at the top is given by the formula:
PE_gravity = m * g * h_gravity
where h_gravity is the vertical distance the skier would have traveled if there were no friction.
To find h_gravity, we need to find the horizontal distance (d_horizontal) traveled along the slope:
d_horizontal = h / sin(θ)
where sin(θ) represents the sine of the angle θ.
Given information:
Angle of the incline (θ) = 18 degrees
Vertical distance (h) = 12.2 m
Step 4: Find the work done by friction.
The work done by friction (W_friction) is given by the formula:
W_friction = -µ * m * g * d_horizontal
where µ is the coefficient of friction.
Step 5: Equate the initial kinetic energy to the final potential energy and the work done by friction.
KE_initial = PE_final + W_friction
Solving this equation will help us find the coefficient of friction (µ).
Now let's put all the values together and solve for the average coefficient of friction (µ).
ke at bottom = (1/2) m v^2
= (1/2) m (144) = 72 m Joules
pe at top = m g h= m (9.81)(12.2 sin 18
= 37 m Joules
so
Lost 72-37 = 35 m Joules to friction
Friction force
= mu m g cos 18= 9.33 mu m
work done by friction
= 9.33 mu m * 12.2 = 114 mu m
so
35 m = 114 mu m
mu = 35/114 = .307
sticky snow :(