A skier skis down a 25 m high frictionless ski slope. How fast will she be moving at the bottom of the slope if:
A) She starts from rest at the top
B) She starts out with a velocity of 10 m/s
KE at top + PE at top=KE at bottom
Can you handle it from here?
To determine the velocity of the skier at the bottom of the slope, we can use the principle of conservation of mechanical energy.
A) If the skier starts from rest at the top of the slope, all of her initial potential energy will be converted into kinetic energy at the bottom of the slope. Since the slope is frictionless, there is no loss of energy due to friction.
The total mechanical energy is given by the sum of the potential energy (PE) and kinetic energy (KE):
E = PE + KE
At the top of the slope, the skier has only potential energy since she is stationary:
E(top) = PE(top) = mgh
Where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the slope (25 m).
At the bottom of the slope, the skier has only kinetic energy since she has converted all of her potential energy into kinetic energy:
E(bottom) = KE(bottom) = 0.5 * m * v²
Where v is the velocity of the skier at the bottom of the slope.
Since mechanical energy is conserved, we can equate the initial potential energy with the final kinetic energy:
E(top) = E(bottom)
mgh = 0.5 * m * v²
Simplifying and solving for v:
v² = 2gh
v = √(2gh)
Plugging in the given values, we have:
v = √(2 * 9.8 * 25)
v ≈ 22 m/s
Therefore, if the skier starts from rest at the top of the slope, she will be moving at approximately 22 m/s at the bottom of the slope.
B) If the skier starts with an initial velocity of 10 m/s, we need to consider both the initial kinetic energy and the change in potential energy during the descent.
Again, using the conservation of mechanical energy:
E(top) = E(bottom)
0.5 * m * (10)² + mgh = 0.5 * m * v²
Simplifying and solving for v:
100 + 25gh = 0.5 * v²
v² = 200 + 50gh
v = √(200 + 50gh)
Plugging in the given values, we have:
v = √(200 + 50 * 9.8 * 25)
v ≈ 36 m/s
Therefore, if the skier starts with an initial velocity of 10 m/s, she will be moving at approximately 36 m/s at the bottom of the slope.