A novice skier, starting from rest, slides down a frictionless 31.0° incline whose vertical height is 123 m. How fast is she going when she reaches the bottom?
Her loss of GPE was mg*123
her gain of KE was 1/2 m v^2
set them equal, solve for v
To find the speed of the skier when they reach the bottom of the incline, we can use the principle of conservation of mechanical energy.
The initial mechanical energy at the top of the incline consists of gravitational potential energy (mgh), where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height of the incline.
The final mechanical energy at the bottom of the incline consists of kinetic energy (0.5mv^2), where v is the velocity of the skier.
Since the incline is frictionless, there is no work done by friction and no mechanical energy loss. Therefore, the initial mechanical energy is equal to the final mechanical energy.
mgh = 0.5mv^2
We can cancel out the mass (m) on both sides of the equation:
gh = 0.5v^2
Solving for v, we get:
v^2 = 2gh
Taking the square root of both sides of the equation:
v = √(2gh)
Now we can substitute the given values into the equation:
g = 9.8 m/s^2
h = 123 m
v = √(2 * 9.8 m/s^2 * 123 m)
Calculating this expression, we get:
v ≈ 39.86 m/s
Therefore, the novice skier is going approximately 39.86 m/s when she reaches the bottom of the incline.
To find the speed of the skier when she reaches the bottom of the incline, we can use the principles of conservation of energy.
First, let's analyze the situation:
- The skier starts from rest, so her initial speed (v₀) is 0 m/s.
- The incline has a vertical height (h) of 123 m.
- The incline is frictionless, so there is no loss of energy due to friction.
We can calculate the final speed (v) of the skier at the bottom of the incline using the conservation of energy equation:
Potential Energy (PE) at the top = Kinetic Energy (KE) at the bottom
The potential energy at the top of the incline can be calculated using the formula: PE = m * g * h
Where:
- m is the mass of the skier (which we'll assume to be 1 kg for simplicity)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- h is the vertical height of the incline (123 m)
So, the potential energy at the top of the incline is: PE = 1 kg * 9.8 m/s² * 123 m = 1205.4 Joules.
And the kinetic energy at the bottom of the incline can be calculated using the formula: KE = (1/2) * m * v²
Since the initial speed (v₀) is 0 m/s, we can set the initial kinetic energy to 0: KE₀ = 0.
Therefore, the conservation of energy equation becomes: PE = KE = KE₀ + KE = m * g * h = (1/2) * m * v²
Now we can solve for the final speed (v) at the bottom of the incline:
m * g * h = (1/2) * m * v²
v² = 2 * g * h
v = sqrt(2 * g * h)
Substituting the known values: v = sqrt(2 * 9.8 m/s² * 123 m)
Calculating the above expression will give us the speed (in m/s) of the skier when she reaches the bottom of the incline.