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 equations of motion for an object on an inclined plane.

Let's break down the problem into smaller steps.

Step 1: Determine the acceleration of the skier on the incline.
The skier experiences a change in velocity from 7.49 m/s to 28.5 m/s while moving down the incline. To find the acceleration, we can use the following formula of motion:

v^2 = u^2 + 2as

Where:
v = final velocity = 28.5 m/s
u = initial velocity = 7.49 m/s
a = acceleration (unknown)
s = distance traveled on the incline (unknown)

Plugging in the values, the equation becomes:

(28.5)^2 = (7.49)^2 + 2a * s ----(1)

Step 2: Determine the acceleration due to gravity component parallel to the incline.
The skier is moving down an incline, so we need to consider the gravitational force component acting parallel to the incline. This component can be calculated using the formula:

A = g * sin θ

Where:
g = acceleration due to gravity = 9.8 m/s^2 (approximate value, assuming Earth's surface)
θ = angle of the incline = 14.4°

Plugging in the values, the equation becomes:

A = (9.8 m/s^2) * sin (14.4°)

Step 3: Substitute the acceleration into equation (1).
Now, we can substitute the value of the acceleration due to gravity component parallel to the incline (A) into equation (1):

(28.5)^2 = (7.49)^2 + 2A * s

Step 4: Solve for the distance traveled on the incline (s).
Rearrange the equation to solve for s:

(28.5)^2 - (7.49)^2 = 2A * s

Substitute the value of A and solve for s:

s = [(28.5)^2 - (7.49)^2] / (2 * (9.8 m/s^2) * sin (14.4°))

Calculating the right side of the equation:

s = [(812.25) - (55.96)] / (19.6 * 0.2486)

s ≈ 44.45 meters

Therefore, the length of the incline is approximately 44.45 meters.

To find the length of the incline, we need to use the principles of conservation of energy.

At the top of the incline, the skier has potential energy due to the elevation and kinetic energy due to the horizontal speed. At the bottom of the incline, the skier has a higher kinetic energy due to the increased speed and potential energy due to the decreased elevation.

Let's break it down step by step:

1. Calculate the change in potential energy:
- The initial potential energy is given by mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the elevation.
- Since the skier starts on a horizontal surface, the initial potential energy is zero (h = 0).

2. Calculate the change in kinetic energy:
- The initial kinetic energy is given by 0.5mv^2, where m is the mass of the skier and v is the initial velocity (7.49 m/s).
- The final kinetic energy is given by 0.5mv^2, where m is the mass of the skier and v is the final velocity (28.5 m/s).

3. Calculate the change in potential energy at the bottom of the incline:
- The final potential energy is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height of the incline.

Since the slope of the incline is given (14.4°), we can calculate the height (h) using trigonometry:
h = l * sin(14.4°),
where l is the length of the incline.

4. Equate the changes in potential and kinetic energy:
- The change in potential energy is equal to the change in kinetic energy:
mgh = 0.5mv^2 - 0.5mu^2,
where m is the mass, g is the acceleration due to gravity, h is the height, v is the final velocity, and u is the initial velocity.

5. Solve for the length of the incline (l):
- Substitute the value of h obtained from step 3 into the equation from step 4 and solve for l:
m * (l * sin(14.4°)) * g = 0.5m * v^2 - 0.5m * u^2.
Notice that the mass cancels out.
Rearrange the equation to solve for l:
l = (0.5 * v^2 - 0.5 * u^2) / (g * sin(14.4°)).

Now, substitute the given values into the equation to find the length of the incline.