A ski jumper glides down a 30.0 degrees slope for 80.0 ft before taking off from a negligibly short horizontal ramp. If the jumpers takeoff speed is 45.0 ft/s, what is the coefficient of kinetic friction between skis and slope?
friction force up slope = mu m g cos 30
work done by friction = -80 mu m g cos 30
drop in height = 80 sin 30 = 40 ft
potential energy drop = m g (40 ft)
ke = (1/2) m v^2
so
(1/2)m(45)^2 = 40 m g - 80 mu m g cos 30
1012 = 40(32) - mu * 32 * 69.3
1012 = 1280 - 2218 mu
mu = (1288 - 1012)/2218 = .124
Ah
To solve this problem, we can use the conservation of energy principle. We'll assume that the skier's initial potential energy is converted into both kinetic energy and work done against friction.
The change in potential energy is given by the formula:
ΔPE = m * g * h
Where:
m is the mass of the skier
g is the acceleration due to gravity (9.8 m/s^2)
h is the vertical height of the slope
Since the slope is inclined at an angle of 30.0 degrees, the vertical height can be calculated as:
h = 80.0 ft * sin(30.0 degrees)
Next, let's determine the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance over which it acts:
Work = force of friction * distance
The force of friction can be calculated using the formula:
force of friction = coefficient of friction * m * g
We want to find the coefficient of kinetic friction, so we rearrange the equation:
coefficient of friction = force of friction / (m * g)
Substituting the known values, we can calculate the coefficient of kinetic friction.
To find the coefficient of kinetic friction between the skis and the slope, you need to use the equations of motion and consider the forces acting on the skier.
Let's break down the problem step by step:
1. First, let's find the acceleration of the skier on the slope. The acceleration can be found using the equation:
a = g * sin(θ)
Where a is the acceleration, g is the acceleration due to gravity (which is approximately 32.2 ft/s^2), and θ is the angle of the slope, which is given as 30.0 degrees.
Plugging in the values, we get:
a = 32.2 ft/s^2 * sin(30.0 degrees)
a ≈ 16.1 ft/s^2
2. Next, we can find the time it takes for the skier to slide down the slope using the equation of motion:
d = v_i * t + (1/2) * a * t^2
Where d is the distance traveled, v_i is the initial speed, a is the acceleration, and t is the time.
We know the distance traveled is 80.0 ft, the initial speed is 0 ft/s (since the skier starts from rest), and the acceleration is 16.1 ft/s^2. Plugging these values into the equation, we can solve for t.
80.0 ft = 0 ft/s * t + (1/2) * 16.1 ft/s^2 * t^2
Rearranging the equation, we get a quadratic equation:
8.05 t^2 = 80.0
t^2 = 80.0 / 8.05
t ≈ 3.13 s
3. Now that we know the time it takes for the skier to slide down the slope, we can calculate the distance traveled horizontally (ramp length) during that time.
Since the skier's takeoff speed is horizontal, we can use the equation:
r = v_i * t
Where r is the distance traveled horizontally, v_i is the initial speed, and t is the time.
Plugging in the values, we get:
r = 45.0 ft/s * 3.13 s
r ≈ 140.9 ft
4. Finally, we can find the coefficient of kinetic friction using the equation:
μ = tan(θ)
Where μ is the coefficient of kinetic friction and θ is the angle of the slope, which is given as 30.0 degrees.
Plugging in the values, we get:
μ = tan(30.0 degrees)
μ ≈ 0.577
Therefore, the coefficient of kinetic friction between the skis and the slope is approximately 0.577.