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 38.3 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?
To determine the height h of the first hill so that the skier just loses contact with the snow at the crest of the second hill, we can use the principle of conservation of energy. At the crest of the second hill, the skier loses contact with the snow, which means their energy is entirely in the form of potential energy.
Let's break down the problem into different stages:
Stage 1: Skier at the top of the first hill
At the top of the first hill, the skier has potential energy (PE) equal to mgh, where m is the mass of the skier, g is the acceleration due to gravity, and h is the height of the first hill.
Stage 2: Skier at the bottom of the first hill
As the skier goes down the first hill, their potential energy is converted to kinetic energy (KE) according to the law of conservation of energy. At the bottom of the first hill, all the potential energy is converted to kinetic energy, so we have:
mgh = (1/2)mv^2, where v is the velocity of the skier at the bottom of the first hill.
Stage 3: Skier at the crest of the second hill
Now, the skier goes up the second hill. At the crest of the second hill, the skier just loses contact with the snow, meaning their velocity becomes zero. The total energy of the skier at the crest is entirely in the form of potential energy:
PE = mgh2, where h2 is the height of the crest of the second hill (what we're trying to find).
Since energy is conserved, we can equate the potential energy at the top of the first hill to the potential energy at the crest of the second hill:
mgh = mgh2
Canceling out the mass, we get:
gh = gh2
Finally, solving for h2, the height of the crest of the second hill, we get:
h2 = h
Therefore, the height h of the first hill must be equal to the height h2 of the crest of the second hill in order for the skier to just lose contact with the snow at the crest of the second hill.