A skier swoops down a hill and over a ramp. She starts from rest at a height of 18m, leaves the ramp at an angle of 45 degrees, and just clears the snowman on her way down, making an angle of 30 degrees with the vertical as she does. Assuming that there is no friction, and that she is small compared to the dimensions of the problem, solve for the height of the snowman in meters.

WORK
So I was thinking you need to do:
Total Energy Before = Total Energy After
Eg= Ek' + mgh'

But I don't know how you can solve for the height of the ramp.... :/

To solve for the height of the ramp, we can break down the motion of the skier into different parts and analyze each part separately.

First, let's determine the skier's velocity as she leaves the ramp. We can use the conservation of energy principle to relate the initial potential energy (mgh) to the final kinetic energy (1/2mv^2).

Eg = Ek'
mgh = 1/2mv^2

The mass of the skier cancels out, and we can solve for the velocity v:

gh = 1/2v^2
v = sqrt(2gh)

Next, let's consider the motion of the skier as she clears the snowman. At this point, we can use the vertical and horizontal components of the velocity to find the height of the snowman.

Since the skier leaves the ramp at an angle of 45 degrees and makes an angle of 30 degrees with the vertical, the vertical component of the velocity would be sin(45) × v and the horizontal component would be cos(45) × v.

As the skier clears the snowman, the vertical displacement is equal to the height of the snowman. We can use the vertical motion equation:

y = y0 + v0y * t - 1/2gt^2

Here, y0 is the initial vertical position (height of the ramp), v0y is the initial vertical velocity, t is the time, and g is the acceleration due to gravity. At the highest point of the trajectory, the vertical velocity is zero, so we have:

0 = v0y - gt

Solving for t, we get:

t = v0y / g

Now, substituting t in the vertical motion equation, and considering the vertical velocity component:

y = y0 + sin(45) * v * (v0y / g) - 1/2g(v0y / g)^2

Since y is the height of the snowman, y0 is 18m (height of the ramp), and v0y can be obtained as sin(30) × v. Substituting these values and simplifying, we can solve for y (height of the snowman).

By following these steps, you should be able to calculate the height of the snowman in meters.