A skier starts at rest and glides (without friction) directly down the fall line from the highest point of a giant snowball with a radius of r=31.7 m . The skier can be observed as a point mass and the air resistance is negligible.
What speed v does the skier have when she loses contact with the snowball?
d = r = 31.7m = Vertical distance
traveled.
V^2 = Vo^2 + 2g*d.
V^2 = 0 + 19.6*31.7 = 621.32,
V = 24.9 m/s.
To determine the speed of the skier when she loses contact with the snowball, we need to use conservation of energy. At the highest point of the snowball, all of the potential energy is converted into kinetic energy.
The potential energy (PE) at the highest point is given by:
PE = mgh
Where:
m is the mass of the skier,
g is the acceleration due to gravity, and
h is the height of the snowball.
Since the skier starts from rest, her initial kinetic energy (KE) is zero.
Therefore, at the highest point, the total mechanical energy (E) is given by:
E = PE + KE = mgh + 0 = mgh
The total mechanical energy at the highest point is equal to the kinetic energy when the skier loses contact with the snowball.
At any point during the skier's descent, the total mechanical energy is given by:
E = 1/2 mv^2
Where:
m is the mass of the skier, and
v is the speed of the skier.
Since no external forces (like friction or air resistance) are acting on the skier, the total mechanical energy is conserved throughout the motion.
Therefore, we can equate the total mechanical energy at the highest point to the total mechanical energy when the skier loses contact with the snowball:
mgh = 1/2 mv^2
From this equation, we can solve for v:
v^2 = 2gh
Taking the square root of both sides:
v = sqrt(2gh)
Substituting g = 9.8 m/s^2 (acceleration due to gravity) and h = r (height of the snowball, which is equal to its radius), we get:
v = sqrt(2 * 9.8 * 31.7)
v ≈ 24.9 m/s
Therefore, the skier has a speed of approximately 24.9 m/s when she loses contact with the snowball.