A skier of mass 55.0 kg slides down a slope 11.7 m long,
inclined at an angle f to the horizontal. The magnitude of
the kinetic friction is 41.5 N. The skier’s initial speed is
65.7 cm/s and the speed at the bottom of the slope is
7.19 m/s. Determine the angle f from the law of conserva-
tion of energy. Air resistance is negligible.
To determine the angle f using the law of conservation of energy, we can set the initial potential energy of the skier equal to the sum of the final kinetic energy and work done by friction.
The initial potential energy (PEi) of the skier is given by:
PEi = mgh
where
m = mass of the skier = 55.0 kg
g = acceleration due to gravity = 9.81 m/s^2
h = vertical height = 11.7 m * sin(f)
The final kinetic energy (KEf) of the skier is given by:
KEf = 0.5 * m * v^2
where
v = speed at the bottom of the slope = 7.19 m/s
The work done by friction is given by:
Wfriction = Ffriction * d
where
Ffriction = magnitude of kinetic friction = 41.5 N
d = length of the slope = 11.7 m
According to the law of conservation of energy:
PEi = KEf + Wfriction
Substitute the equations for PEi, KEf, and Wfriction into the conservation of energy equation, and solve for f.
mgh = 0.5 * m * v^2 + Ffriction * d
55.0 kg * 9.81 m/s^2 * 11.7 m * sin(f) = 0.5 * 55.0 kg * (7.19 m/s)^2 + 41.5 N * 11.7 m
5826.69 * sin(f) = 2241.22 + 484.05
5826.69 * sin(f) = 2725.27
sin(f) = 2725.27 / 5826.69
sin(f) = 0.467971
f = arcsin(0.467971)
f ≈ 28.1 degrees
Therefore, the angle f is approximately 28.1 degrees.