A skier is gliding along at 1.13 m/s on horizontal, frictionless snow. He suddenly starts down a 13.1° incline. His speed at the bottom is 18.8 m/s. What is the length of the incline?

Use his change in kinetic energy to get his loss of potential energy. The equation

g H = change in (V^2)/2
can be used to het the height change, H.

H and the slope angle can be used to get the length L of the incline.

H/L = sin 13.1

To find the length of the incline, we can apply the principles of kinematics. One way to approach this problem is by using the equations of motion.

Let's assume that the length of the incline is represented by 'd'.

We know that the skier starts with an initial speed of 1.13 m/s on the horizontal surface. Since there is no friction, this horizontal speed remains constant throughout the motion.

When the skier starts moving down the incline, the constant horizontal speed does not change. However, the skier's motion now has a vertical component due to the incline.

Using the given angle of inclination (13.1°), we can calculate the component of the skier's initial horizontal speed that acts parallel to the incline.

The parallel component of velocity (Vp) is given by Vp = V * cos(θ), where V is the magnitude of the initial velocity (1.13 m/s) and θ is the angle of inclination (13.1°).

Vp = 1.13 m/s * cos(13.1°)
Vp ≈ 1.13 m/s * 0.970
Vp ≈ 1.098 m/s

Now, we can determine the time it takes for the skier to travel down the incline. We can use the equation of motion for projectile motion in the vertical direction:

Vf = Vi + gt

Where Vf is the final vertical velocity (0 m/s at the bottom), Vi is the initial vertical velocity (0 m/s), g is the acceleration due to gravity (9.8 m/s²), and t is the time taken.

0 m/s = 0 m/s + (9.8 m/s²) * t

Solving for t:

t = 0 / (9.8 m/s²)
t = 0 seconds

Since the vertical displacement is zero, the skier reaches the bottom instantaneously.

Next, we can relate the horizontal distance traveled (d) to the parallel component of velocity (Vp) and the time (t):

d = Vp * t

d = 1.098 m/s * 0 seconds
d = 0 meters

Therefore, the length of the incline is 0 meters.

Note: It's important to analyze the given information and use the appropriate equations to solve the problem. Always double-check your calculations and consider if the solution aligns with reality.