A skier is traveling at a speed of 39.7 m/s when she reaches the base of a frictionless ski hill. This hill makes an angle of 14° with the horizontal. She then coasts up the hill as far as possible. What height (measured vertically above the base of the hill) does she reach?
Since all her KE is converted to PE,
1/2 mv^2 = mgh
so,
1/2 * 39.7^2 = 9.81h
Note that the angle of incline makes no difference.
To find the height that the skier reaches, we can use the principles of conservation of energy and trigonometry.
Step 1: Convert the given speed to vertical and horizontal components.
The vertical velocity component (V_y) can be found using the formula V_y = V * sin(theta), where V is the initial speed and theta is the angle of the hill with the horizontal.
V_y = 39.7 m/s * sin(14°)
V_y ≈ 9.63 m/s
The horizontal velocity component (V_x) can be found using the formula V_x = V * cos(theta), where V is the initial speed and theta is the angle of the hill with the horizontal.
V_x = 39.7 m/s * cos(14°)
V_x ≈ 38.04 m/s
Step 2: Calculate the skier's initial potential energy at the base of the hill.
The initial potential energy (PE_base) can be calculated using the formula PE_base = m * g * h, 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 above the base.
PE_base = m * g * h_base
Step 3: Calculate the skier's final potential energy at the maximum height.
The final potential energy (PE_max) can also be calculated using the formula PE_max = m * g * h_max, where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h_max is the vertical height achieved by the skier.
PE_max = m * g * h_max
Step 4: Equate the initial and final potential energies to find the maximum height.
Since the ski hill is frictionless, the skier's initial kinetic energy is transformed entirely into potential energy at the maximum height. This means that PE_base = PE_max.
m * g * h_base = m * g * h_max
Step 5: Simplify and solve for h_max.
Dividing both sides of the equation by m * g, we get:
h_max = h_base
Therefore, the skier reaches the same height vertically above the base of the hill as she started. The height reached is equal to the vertical distance between the base of the hill and the starting position of the skier.
To find the height the skier reaches, we can apply the principle of conservation of mechanical energy. The mechanical energy of the skier is conserved as there is no friction and air resistance. The total mechanical energy is the sum of the kinetic energy and the potential energy.
1. Start by calculating the initial kinetic energy of the skier. The kinetic energy (KE) is given by the formula:
KE = (1/2)mv^2
Where m is the mass of the skier and v is their velocity. Since the mass is not given in the problem, we can assume it cancels out.
KE = (1/2)(39.7)^2 = 0.5 * 39.7^2 = 789.405 J (Joules)
2. Next, calculate the potential energy (PE) at the highest point the skier reaches. The formula for potential energy is:
PE = 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 height.
3. To eliminate the mass from the equation, we can divide both sides of the equation by m:
PE/m = gh
4. Now, we need to find h. To determine the vertical height the skier reaches, we need to consider the angle of the ski hill and using trigonometry.
We can calculate the vertical height (h) using the formula:
h = d * sin(θ)
Where d is the distance along the slope of the hill and θ is the angle of the hill (14°).
5. To find the distance (d), we can use the formula:
d = v * t
Where v is the initial velocity of the skier and t is the time required for the skier to reach the highest point. Note that the time taken to reach the maximum height is the same as the time taken to reach the base of the hill since the hill is frictionless.
6. Rearrange the formulas to solve for t:
t = d / v
7. Substitute the value of v and calculate d:
d = v * sin(θ)
d = 39.7 * sin(14°)
8. Substitute the value of d into the equation to solve for h:
h = d * sin(θ)
9. Finally, calculate the potential energy at the highest point:
PE = mgh
Now, we have all the calculations and can obtain the final answer for the height (h) the skier reaches above the base of the hill.