A skier is gliding along at 5.0m/s on horizontal, frictionless snow. He suddenly starts down a 13 degrees incline. His speed at the bottom is 12m/s.

What is the length of the incline?
How long does it take him to reach the bottom?

Conservation of energy and the speed at the bottom can be used to determine the vertical drop. Call it H. Calculate it.

The length of the incline is H/sin13.

Time to reach the bottom is
T = (length of incline)/(average speed)
= H/[(sin13)*6m/s)

To find the length of the incline, we can use the concept of conservation of energy. When the skier is at the top, he has potential energy due to his height above the ground, and this potential energy is converted into kinetic energy as he moves down the incline.

The potential energy (PE) of an object at a height h is given by: PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and h is the height.

The kinetic energy (KE) of an object moving with velocity v is given by: KE = (1/2)mv².

Assuming the skier's mass is m, we can equate the potential energy at the top of the incline to the kinetic energy at the bottom of the incline:

mgh = (1/2)mv².

Since the mass of the skier cancels out, we have:

gh = (1/2)v².

Using the given values: h = 0 (since the incline starts at ground level), g = 9.8 m/s², and v = 12 m/s, we can solve for h:

9.8h = (1/2)(12)².
9.8h = 72.
h = 72/9.8.

By using the equation h = l*sinθ, where l is the length of the incline and θ is the angle of the incline, we can find l:

l*sin(13°) = 72/9.8.
l*sin(13°) = 7.347.

Dividing both sides by sin(13°), we get:

l = 7.347/sin(13°).
l ≈ 33.63 meters.

Therefore, the length of the incline is approximately 33.63 meters.

To find the time it takes for the skier to reach the bottom, we can use the concept of acceleration due to gravity. The acceleration (a) of the skier down the incline is given by a = g * sin(θ), where g is the acceleration due to gravity and θ is the angle of the incline.

Using the given values of θ = 13° and g = 9.8 m/s², we can find a:

a = 9.8 * sin(13°).
a ≈ 2.113 m/s².

To find the time (t) it takes for the skier to reach the bottom, we can use the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.

Using the given values of u = 5.0 m/s, v = 12 m/s, and a = 2.113 m/s², we can rearrange the equation to solve for t:

t = (v - u) / a.
t = (12 - 5.0) / 2.113.
t ≈ 3.32 seconds.

Therefore, it takes approximately 3.32 seconds for the skier to reach the bottom of the incline.