Why Does a Skier Go Airborne at h=R/3 on a Hemispherical Hill?

[PhysicsinCalifornia]

I need help on a physics problem I've been working on...

A skier starts at rest at the top of a large hemispherical hill with radius (height) = R. Neglecting friction, show that the skier will leave the hill and become airborne at the distance of h = R/3 below the top of the hill.

I understand that at the point the skier goes airborne, the normal force is zero, but how do I conceptually show it? When it's at the crest of the hill, there are obviously two vertical forces in play: the weight of the skier (downward) and the normal force (upward).

So the question, once again, is how do I show specifically (to prove) that the skier goes airborne at height = 3/R.

Thanks in advance.

 

[jdavel]

Physics,

Can you find an equation for the normal force as a function of the skier's angular position, where angle is measured from the vertical. In other words he starts at theta=0 and flies off somewhere between theta=0 and theta=90degs? You need an equation for the normal force in terms of theta.

 

[boaz]

like it was said by jdavel, you need to know the angle. if you do a "free body diagram", you will notice that :
$$mg \cdot \cos \theta - N = m \cdot \frac{V^2}{R} \rightarrow \cos \theta = \frac{ \frac{V^2}{R} + N}{g}$$

 

[Doc Al]

more hints

Since you are asked to find the point of departure in terms of height h below the hilltop, rewrite $\cos\theta$ in terms of h and R. Hint: You'll need to use conservation of energy.

 

[PhysicsinCalifornia]

I need help on a physics problem I've been working on...

A skier starts at rest at the top of a large hemispherical hill with radius (height) = R. Neglecting friction, show that the skier will leave the hill and become airborne at the distance of h = R/3 below the top of the hill.

I understand that at the point the skier goes airborne, the normal force is zero, but how do I conceptually show it? When it's at the crest of the hill, there are obviously two vertical forces in play: the weight of the skier (downward) and the normal force (upward).

So the question, once again, is how do I show specifically (to prove) that the skier goes airborne at height = 3/R.

Thanks in advance.

*Correction**

The skier goes airborne at height h= R/3