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 kinematics. Here's how you can solve it step by step:
Step 1: Determine the acceleration of the skier.
Since the incline is at an angle to the horizontal, the component of the skier's weight parallel to the incline will act as the driving force down the incline. The formula for the force down the incline can be calculated by multiplying the skier's weight (mg) by the sine of the angle of incline (θ).
So, the formula for the force down the incline is: F = mg * sin(θ)
Where:
- F is the force down the incline
- m is the mass of the skier
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- θ is the angle of incline (14.4°)
Step 2: Calculate the acceleration.
The force down the incline (F) is equal to the mass of the skier (m) multiplied by the acceleration (a). Therefore: F = ma
Substituting the formula for the force down the incline into the equation, we get:
mg * sin(θ) = ma
Canceling out the mass (m) on both sides, we find:
g * sin(θ) = a
Now, substitute the values into the equation:
a = (9.8 m/s²) * sin(14.4°)
Calculate it and you will get the acceleration value.
Step 3: Find the time taken to reach the bottom.
To find the time taken, we can use the kinematic equation:
v = u + at
Where:
- v is the final velocity at the bottom (28.5 m/s)
- u is the initial velocity on the horizontal (7.49 m/s)
- a is the acceleration calculated in step 2
- t is the time taken
Rearranging the equation to solve for time (t), we get:
t = (v - u) / a
Substitute the known values into the equation and calculate the time taken.
Step 4: Calculate the length of the incline.
The length of the incline can be calculated using the formula:
s = ut + (1/2)at²
Where:
- s is the distance or length of the incline
- u is the initial velocity on the horizontal (7.49 m/s)
- t is the time taken calculated in step 3
- a is the acceleration calculated in step 2
Substitute the known values into the equation and solve for distance (s).
After completing these four steps, you should be able to find the length of the incline.