Conservation of energy of a downhill skier

In summary, A skier starting from rest at the top of a hill coasts down and up a second hill with a circular crest of radius r = 47m. Neglecting friction and air resistance, the height h of the first hill must be equal to the radius of the second hill in order for the skier to just lose contact with the snow at the crest of the second hill. This can be determined by using Newton's 2nd law to find the required speed of the skier at the top of the second hill, and then using that speed to calculate the height of the starting point.
  • #1
spidey12
37
0

Homework Statement



A skier starts from rest at the top of a hill. The skier coasts down the hill and up a second hill, as the drawing illustrates. The crest of the second hill is circular, with a radius of r = 47 m. Neglect friction and air resistance. What must be the height h of the first hill so that the skier just loses contact with the snow at the crest of the second hill?

http://img254.imageshack.us/img254/7408/0644bg0.gif

Homework Equations



1/2mvf2+mgHf=1/2mvo2+mgho


The Attempt at a Solution



All the masses cancel out. I got the equation down to two unknowns, one is the initial height and the other is the final velocity. Now I'm stumped.

This is the furthest I've gotten: 1/2vf2+g(47+h)=1/2vo2+g(47-h)
 
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  • #2
You'll need more than conservation of energy. (Hints: Measure PE from the dashed line. What's the initial speed?)

Figure out the speed that the skier must have at the top of the second hill to just lose contact. Hint: Newton's 2nd law.
 
  • #3
Doc Al said:
You'll need more than conservation of energy. (Hints: Measure PE from the dashed line. What's the initial speed?)

Figure out the speed that the skier must have at the top of the second hill to just lose contact. Hint: Newton's 2nd law.

Ive been staring at this but i can't seem to squeeze a formula out of it..can you be a little more specifc..
 
  • #4
Another hint: What kind of motion must the skier execute when climbing the second hill? What's his acceleration?
 
  • #5
Doc Al said:
Another hint: What kind of motion must the skier execute when climbing the second hill? What's his acceleration?

in going from the lowest point on to the top of the crest the skier is losing kinetic energy and gaining potential energy.

Also in reference to your Newton's second law hint, how does the energy-work theorm fit into this: Wnc=Ef-E0
 
  • #6
spidey12 said:
in going from the lowest point on to the top of the crest the skier is losing kinetic energy and gaining potential energy.
That's true, of course, but that hint about the "kind of motion" had to do with Newton's 2nd law, not energy conservation.
Also in reference to your Newton's second law hint, how does the energy-work theorm fit into this: Wnc=Ef-E0
It doesn't.

What's Newton's 2nd law? Consider the shape of the second hill.
 
  • #7
Doc Al said:
That's true, of course, but that hint about the "kind of motion" had to do with Newton's 2nd law, not energy conservation.

It doesn't.

What's Newton's 2nd law? Consider the shape of the second hill.

Newtons second law is F=ma. and the second crest is a circle with a radius of 47m. Am i now trying to find the centripedal acceleration? I think I'm way off.
 
  • #8
spidey12 said:
Newtons second law is F=ma. and the second crest is a circle with a radius of 47m.
Good.
Am i now trying to find the centripedal acceleration?
Yes!
I think I'm way off.
You are finally on track. :wink:
 
  • #9
Doc Al said:
Good.

Yes!

You are finally on track. :wink:



Ok, how can I find centripedal acceleration if i do not have the velocity? Ac=v2/r
 
  • #10
spidey12 said:
Ok, how can I find centripedal acceleration if i do not have the velocity? Ac=v2/r
Using Newton's 2nd law! Realize that your job is to find the speed required at the top of the second hill, then use that to figure out the height h of the starting point.
 
  • #11
Doc Al said:
Using Newton's 2nd law! Realize that your job is to find the speed required at the top of the second hill, then use that to figure out the height h of the starting point.

Aha! I got it. Thank you doc al, you're help is sooooo greatly appreciated.
 
  • #12
Doc Al said:
Using Newton's 2nd law! Realize that your job is to find the speed required at the top of the second hill, then use that to figure out the height h of the starting point.

Wait what would the Forces be in the F=ma then? the acceleration would be the centripetal acceleration and the m would cancel I am guessing but what are the forces?
 

Related to Conservation of energy of a downhill skier

1. How does the conservation of energy apply to a downhill skier?

The law of conservation of energy states that energy cannot be created or destroyed, only transferred from one form to another. This applies to a downhill skier as they convert potential energy (from being at a higher elevation) into kinetic energy (movement) as they ski downhill.

2. Are there any factors that affect the conservation of energy for a downhill skier?

Yes, there are several factors that can affect the conservation of energy for a downhill skier. These include the slope of the hill, the weight of the skier, air resistance, and the type of snow they are skiing on.

3. How does the conservation of energy impact a downhill skier's speed?

According to the law of conservation of energy, the total energy of a system remains constant. This means that as a downhill skier's potential energy decreases, their kinetic energy (and therefore their speed) will increase, until they reach the bottom of the hill where all of their potential energy has been converted into kinetic energy.

4. How can a downhill skier conserve energy while skiing?

A downhill skier can conserve energy by using proper technique, such as making smooth turns and avoiding unnecessary movements. They can also choose to ski on terrain with a lower slope, which will require less energy expenditure.

5. What are the potential consequences of not conserving energy while skiing downhill?

If a downhill skier does not conserve energy, they may become fatigued more quickly and their performance may suffer. Additionally, not conserving energy could lead to a higher risk of accidents or injuries due to lack of control or control over their speed.

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