A skier is gliding along at 6.24 m/s on horizontal, frictionless snow. He suddenly starts down a 19.9° incline. His speed at the bottom is 23.3 m/s. What is the length of the incline?
To find the length of the incline, we can use the conservation of mechanical energy. At the top of the incline, the skier has only kinetic energy, and at the bottom of the incline, the skier has both kinetic and potential energy.
The initial kinetic energy (KE1) is given by:
KE1 = 1/2 * m * v1^2
The final kinetic energy (KE2) is given by:
KE2 = 1/2 * m * v2^2
The change in potential energy (PE) is given by:
PE = m * g * h
Since the incline is frictionless, there is no work done by friction.
According to the conservation of mechanical energy, the initial kinetic energy plus the initial potential energy should be equal to the final kinetic energy:
KE1 + PE = KE2
Since the incline is at an angle of 19.9°, we can write:
h = L * sin(19.9°)
Plugging in the given values:
v1 = 6.24 m/s
v2 = 23.3 m/s
h = L * sin(19.9°)
g = 9.8 m/s^2
We can now solve for L by rearranging the equation and substituting the values:
L = (KE2 - KE1) / (m * g * sin(19.9°))
L = (1/2 * m * v2^2 - 1/2 * m * v1^2) / (m * g * sin(19.9°))
Simplifying further:
L = ((v2^2 - v1^2) / (2 * g * sin(19.9°)))
Plugging in the given values:
L = ((23.3^2 - 6.24^2) / (2 * 9.8 * sin(19.9°)))
Calculating L:
L ≈ 207 meters
Therefore, the length of the incline is approximately 207 meters.
To find the length of the incline, we can start by analyzing the initial and final speeds of the skier and the angle of the incline.
1. We are given the initial speed of the skier on the horizontal snow, which is 6.24 m/s.
2. The skier starts down a 19.9° incline and reaches a final speed of 23.3 m/s at the bottom.
3. We need to find the length of the incline.
To solve this problem, we can use the principles of kinematics and trigonometry.
First, let's break down the initial velocity into its components. The vertical component of the velocity is given by:
v₁_y = v₁ * sin(θ)
where v₁ is the initial speed (6.24 m/s) and θ is the angle of the incline (19.9°).
v₁_y = 6.24 m/s * sin(19.9°)
v₁_y = 2.12 m/s
The horizontal component of the velocity remains unchanged:
v₁_x = v₁ * cos(θ)
v₁_x = 6.24 m/s * cos(19.9°)
v₁_x = 5.88 m/s
Now, let's analyze the final speed at the bottom of the incline. We know the final speed is 23.3 m/s, which is the resultant velocity, v₂. We can break down this velocity into its vertical and horizontal components.
v₂_y = v₂ * sin(α)
where v₂ is the final speed (23.3 m/s) and α is the angle of the incline (19.9°).
v₂_y = 23.3 m/s * sin(19.9°)
v₂_y = 7.88 m/s
v₂_x = v₂ * cos(α)
v₂_x = 23.3 m/s * cos(19.9°)
v₂_x = 22.02 m/s
Next, we can find the change in velocity along the incline using the following equation:
Δv_y = v₂_y - v₁_y
Δv_y = 7.88 m/s - 2.12 m/s
Δv_y = 5.76 m/s
Now, we can calculate the acceleration along the incline using the equation:
a_y = Δv_y / t
where t is the time it takes to reach the bottom of the incline.
a_y = 5.76 m/s / t
Since the slope is frictionless, we can assume there is no acceleration in the horizontal direction (a_x = 0).
Now, using the equations of motion, we can relate the acceleration, time, and distance.
y = y₀ + v₀_y * t + (1/2) * a_y * t²
where y is the vertical distance traveled along the incline, y₀ is the initial vertical position (0 m), and v₀_y is the initial vertical velocity (2.12 m/s).
Setting y = 0 (since the vertical displacement at the bottom is zero), we can rearrange the equation:
0 = 0 + 2.12 m/s * t + (1/2) * a_y * t²
Substituting the expression for a_y and solving for t:
0 = 2.12 m/s * t + (1/2) * (5.76 m/s) * t²
We can solve this quadratic equation to find the time t.
2.88 t² + 2.12 t = 0
By factoring out t:
t(2.88 t + 2.12) = 0
Since time cannot be negative, we discard the solution t = 0.
Solving for t:
2.88 t + 2.12 = 0
2.88 t = -2.12
t = -2.12 / 2.88
t ≈ -0.736 seconds
We discard the negative solution since time cannot be negative. So, the time taken to reach the bottom of the incline is approximately 0.736 seconds.
Finally, we can find the length of the incline using the equation:
l = v₁_x * t
where v₁_x is the initial horizontal velocity (5.88 m/s) and t is the time (0.736 seconds).
l = 5.88 m/s * 0.736 s
l ≈ 4.32 meters
Therefore, the length of the incline is approximately 4.32 meters.