An extreme skier, starting from rest, coasts down a mountain that makes an angle of 26.7 ° with the horizontal. The coefficient of kinetic friction between her skis and the snow is 0.248. She coasts for a distance of 10.5 m before coming to the edge of a cliff. Without slowing down, she skis off the cliff and lands downhill at a point whose vertical distance is 4.88 m below the edge. How fast is she going just before she lands?
To find the speed of the skier just before she lands, we can use the principle of conservation of mechanical energy, taking into account the work done by friction.
First, let's find the speed of the skier at the edge of the cliff. The initial energy of the skier is purely potential energy, given by:
Ep = m * g * h
where m is the mass of the skier, g is the acceleration due to gravity (9.8 m/s²), and h is the height of the cliff (4.88 m). We can assume that the potential energy at the bottom of the cliff is zero.
The final energy of the skier just before she lands includes both potential and kinetic energy:
Ef = m * g * h + (1/2) * m * v²
where v is the velocity of the skier just before she lands.
Since there is no loss or gain of energy due to external work, we have:
Ef = Ep
m * g * h + (1/2) * m * v² = m * g * h
We can cancel out the mass, m, on both sides of the equation:
g * h + (1/2) * v² = g * h
Simplifying the equation:
(1/2) * v² = 0
Now, let's analyze the skier's motion down the mountain. The only force acting on the skier is the force of friction. The force of friction can be calculated using the formula:
f = μ * N
where μ is the coefficient of kinetic friction (0.248) and N is the normal force.
The normal force is equal to the component of the skier's weight perpendicular to the slope, given by:
N = m * g * cos(θ)
where θ is the angle of the mountain with respect to the horizontal (26.7°) and g is the acceleration due to gravity.
The force of friction, f, is equal to the gravitational force acting parallel to the slope. Therefore:
f = m * g * sin(θ)
Using the equation for the force of friction, we can determine the work done by friction as the skier coasts for a distance of 10.5 m:
Wf = f * d
where d is the distance the skier coasts (10.5 m).
Since the work done by friction is equal to the change in mechanical energy, we have:
Wf = Ef - Ep
Substituting the expressions for the work done by friction and the mechanical energies:
m * g * sin(θ) * d = m * g * h
We can cancel out the mass, m, on both sides of the equation:
g * sin(θ) * d = g * h
Substituting the given values:
(9.8 m/s²) * sin(26.7°) * (10.5 m) = (9.8 m/s²) * (4.88 m)
Simplifying the equation:
3.124 * 10.5 m = 9.8 m * (4.88 m)
32.79 m = 47.82 m
Now we have the value for v in our equation:
(1/2) * v² = 0
Therefore, the speed of the skier just before she lands is 0 m/s.