A skier is gliding along at 7.49 m/s on horizontal, frictionless snow. He suddenly starts down a 14.4° incline. His speed at the bottom is 28.5 m/s. What is the length of the incline?
To find the length of the incline, we can use the principles of physics, specifically the conservation of mechanical energy.
Let's break down the problem into steps:
Step 1: Find the vertical component of the initial velocity.
The vertical component of the initial velocity can be found using the formula:
Vertical velocity = Initial velocity * sin(angle)
Given: initial velocity = 7.49 m/s and angle = 14.4°
Vertical velocity = 7.49 m/s * sin(14.4°)
Step 2: Find the time it takes to reach the bottom.
We can use the kinematic equation to find the time taken to reach the bottom:
Final velocity = Initial velocity + Acceleration * time
As the skier moves down the incline, the acceleration is due to gravity acting along the incline. So,
Final velocity = 28.5 m/s and Initial velocity = 7.49 m/s
Acceleration = g * sin(angle)
Here, g represents the acceleration due to gravity (approximately 9.8 m/s²).
Step 3: Find the distance traveled.
The distance traveled down the incline can be found using the equation:
Distance = Average velocity * time
Since the acceleration is constant, the average velocity is given by:
Average velocity = (Initial velocity + Final velocity) / 2
Now, let's calculate the values step by step:
Step 1:
Vertical velocity = 7.49 m/s * sin(14.4°) = 2.073 m/s
Step 2:
Using the equation Final velocity = Initial velocity + Acceleration * time, we have:
28.5 m/s = 7.49 m/s + (9.8 m/s² * sin(14.4°)) * time
Solving for time:
time = (28.5 m/s - 7.49 m/s) / (9.8 m/s² * sin(14.4°))
Step 3:
Calculating the average velocity:
Average velocity = (7.49 m/s + 28.5 m/s) / 2
Now, we can calculate the distance traveled down the incline:
Distance = Average velocity * time
Plug in the values into the equation and calculate to find the length of the incline.
To find the length of the incline, we can use the concepts of conservation of energy and work-energy principle.
First, let's find the change in gravitational potential energy (ΔPE) as the skier moves down the incline. The change in potential energy can be calculated using the formula:
ΔPE = m * g * h
where m represents the skier's mass, g is the acceleration due to gravity, and h is the vertical height or distance the skier travels downhill.
We can set up an equation for the change in potential energy:
ΔPE = PE_final - PE_initial
Since the skier starts at the top of the incline with a speed of 7.49 m/s and reaches the bottom with a speed of 28.5 m/s, we can calculate the initial and final potential energies using the equation:
PE = 1/2 * m * v^2
where v represents the speed of the skier.
Initially,
PE_initial = 1/2 * m * (7.49 m/s)^2
At the bottom of the incline,
PE_final = 1/2 * m * (28.5 m/s)^2
Now we have the equation for the change in potential energy:
ΔPE = PE_final - PE_initial
Next, we need to find the change in kinetic energy (ΔKE) of the skier. Since the slope is frictionless, the change in kinetic energy is simply given by:
ΔKE = KE_final - KE_initial
Initially,
KE_initial = 1/2 * m * (7.49 m/s)^2
At the bottom of the incline,
KE_final = 1/2 * m * (28.5 m/s)^2
Now we have the equation for the change in kinetic energy:
ΔKE = KE_final - KE_initial
According to the work-energy principle, the change in kinetic energy (ΔKE) should be equal to the change in potential energy (ΔPE) when considering only conservative forces:
ΔKE = ΔPE
So we can equate the two equations:
ΔPE = ΔKE
Now we can solve for the height of the incline (h). Rearranging the equation, we get:
m * g * h = 1/2 * m * (28.5 m/s)^2 - 1/2 * m * (7.49 m/s)^2
Mass cancels out from both sides of the equation:
g * h = 1/2 * (28.5 m/s)^2 - 1/2 * (7.49 m/s)^2
Substituting the given values for g, h, and the known constants:
9.8 m/s^2 * h = 1/2 * (28.5 m/s)^2 - 1/2 * (7.49 m/s)^2
Simplifying further:
9.8 m/s^2 * h = 429.8175 m^2/s^2 - 27.964975 m^2/s^2
9.8 m/s^2 * h = 401.852525 m^2/s^2
Finally, we can solve for h:
h = (401.852525 m^2/s^2) / (9.8 m/s^2)
h ≈ 41.01 m
The length of the incline is equal to the hypotenuse of the right triangle formed by the incline and the vertical height. We can calculate the length (L) using the formula:
L = h / sin(θ)
where θ is the angle of the incline.
Substituting the given values:
L = 41.01 m / sin(14.4°)
L ≈ 149.69 m
Therefore, the length of the incline is approximately 149.69 meters.