A 145 m long ramp is to be built for a ski jump. If a skier starting from rest at the top is to have a speed no faster than 22 m/s at the bottom, what should be the maximum angle of inclination?
To find the maximum angle of inclination, we can use the principles of conservation of energy.
The potential energy at the top of the ramp will be equal to the kinetic energy at the bottom of the ramp.
Potential energy at the top = mgh
Kinetic energy at the bottom = (1/2)mv^2
Given:
Length of the ramp (L) = 145 m
Maximum speed at the bottom (v) = 22 m/s
Acceleration due to gravity (g) = 9.8 m/s^2
We can assume the mass (m) of the skier cancels out in the equation, so we don't need to consider it.
Potential energy at the top = Kinetic energy at the bottom
mgh = (1/2)mv^2
Canceling out the mass:
gh = (1/2)v^2
Substituting the known values:
(9.8)(h) = (1/2)(22)^2
Simplifying:
9.8h = 242
Dividing both sides by 9.8:
h = 24.69
Now, we can calculate the maximum angle of inclination using the formula:
tan(theta) = opposite/adjacent
tan(theta) = h/L
Substituting the known values:
tan(theta) = 24.69/145
Calculating theta by taking the inverse tangent:
theta = arctan(24.69/145)
Calculating theta using a calculator, we get:
theta ≈ 9.66 degrees
Therefore, the maximum angle of inclination required for the ramp is approximately 9.66 degrees.
To determine the maximum angle of inclination for the ramp, we can use the principle of conservation of energy. We will equate the initial potential energy of the skier at the top of the ramp to the final kinetic energy at the bottom.
1. First, let's calculate the initial potential energy (PE) at the top of the ramp. The potential energy is given by the formula: PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the vertical height.
2. Next, let's calculate the final kinetic energy (KE) at the bottom of the ramp. The kinetic energy is given by the formula: KE = 0.5 * m * v^2, where m is the mass and v is the velocity.
3. We'll assume that the mass cancels out and solve for the height (h) in terms of the velocity (v). Rearrange the equations: PE = KE --> m * g * h = 0.5 * m * v^2.
4. Cancel out the mass (m) on both sides of the equation: g * h = 0.5 * v^2.
5. Rearrange the equation to solve for the height (h): h = (0.5 * v^2) / g.
6. Substitute the given values into the equation. The velocity is 22 m/s, and the acceleration due to gravity is approximately 9.8 m/s^2.
h = (0.5 * (22 m/s)^2) / 9.8 m/s^2.
Calculating this expression will give you the height (h) of the ramp.
7. Finally, using the height (h) and the length of the ramp (l), we can determine the maximum angle (θ) of inclination using trigonometry. The formula for the maximum angle is: θ = arctan(h / l).
So, by following these steps, you can calculate the maximum angle of inclination for the ramp.