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
see other post.
To find the speed of the skier at the bottom of the slope, we can use the principle of conservation of energy. The total mechanical energy of the system remains constant, which means that the initial potential energy at the top of the slope is converted into kinetic energy at the bottom of the slope.
A) When the skier starts from rest at the top, she has no initial kinetic energy. Therefore, all her initial potential energy will be converted into kinetic energy at the bottom of the slope.
The potential energy (PE) of an object at a certain height is given by the formula:
PE = mgh
where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
In this case, the skier's initial potential energy will be:
PE = mgh = (mass of skier) * 9.8 * 25
Since we know that all this potential energy will be converted into kinetic energy, we can write:
PE = KE = (1/2) * m * v^2
where KE is the kinetic energy of the skier at the bottom of the slope and v is her velocity.
By equating the two expressions for the potential energy and solving for v, we can find the velocity:
(1/2) * m * v^2 = (mass of skier) * 9.8 * 25
Simplifying and solving for v:
v^2 = 2 * 9.8 * 25
v = √(2 * 9.8 * 25)
Therefore, the skier will be moving at the speed of approximately √(2 * 9.8 * 25) m/s at the bottom of the slope when starting from rest at the top.
B) When the skier starts out with an initial velocity of 10 m/s, we need to consider both the initial kinetic energy and the potential energy.
The total initial energy will be the sum of the initial kinetic energy (KE) and the initial potential energy (PE).
KE = (1/2) * m * v_initial^2
PE = mgh
At the bottom of the slope, all this energy will be in the form of kinetic energy:
KE = (1/2) * m * v_final^2
To find the final velocity (v_final), we can equate the initial and final energies:
(1/2) * m * v_initial^2 + mgh = (1/2) * m * v_final^2
Since the mass (m) of the skier appears on both sides of the equation, we can cancel it out:
(1/2) * v_initial^2 + gh = (1/2) * v_final^2
Substituting the given values:
(1/2) * 10^2 + 9.8 * 25 = (1/2) * v_final^2
Simplifying and solving for v_final:
v_final^2 = 100 + 245
v_final = √(100 + 245)
Therefore, the skier will be moving approximately at the speed of √(100 + 245) m/s at the bottom of the slope when starting with an initial velocity of 10 m/s.