A skier is gliding along at 2.62 m/s on horizontal, frictionless snow. He suddenly starts down a 16.9° incline. His speed at the bottom is 18.4 m/s. What is the length of the incline?

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To find the length of the incline, we can use the principle of conservation of energy. The total mechanical energy of the skier remains constant throughout the motion, assuming negligible air resistance and friction.

The initial kinetic energy (KE₁) of the skier is given by ½mv₁², where m is the mass of the skier and v₁ is the initial speed. Since the mass of the skier is not provided, we can assume it cancels out in the equation for energy conservation.

The final kinetic energy (KE₂) of the skier is given by ½mv₂², where v₂ is the final speed.

The initial potential energy (PE₁) of the skier is given by mgh₁, where h₁ is the initial height of the skier.

The final potential energy (PE₂) of the skier is given by mgh₂, where h₂ is the final height of the skier.

Since the incline is frictionless, there is no change in mechanical energy from work done by friction.

Using the equation for conservation of energy:
KE₁ + PE₁ = KE₂ + PE₂

The initial potential energy is converted into kinetic energy as the skier goes down the incline:
0 + mgh₁ = ½mv₂² + mgh₂

Since the mass cancels out, we can simplify the equation to:
gh₁ = ½v₂² + gh₂

Rearranging the equation:
gh₁ - gh₂ = ½v₂²

Factoring out gravity (g):
g(h₁ - h₂) = ½v₂²

We know the angle of the incline (θ) = 16.9°. The vertical height difference (h₂ - h₁) is given by the formula:
(h₂ - h₁) = L * sin(θ), where L is the length of the incline.

Substituting it into the previous equation:
gL * sin(θ) = ½v₂²

Now we can solve for L:
L = (½v₂²) / (g * sin(θ))

Plugging in the given values:
L = (½ * (18.4 m/s)²) / (9.8 m/s² * sin(16.9°))

Calculating the length of the incline:
L ≈ 150 meters

Therefore, the length of the incline is approximately 150 meters.