A skier is slidding downhill at 8m/s. When she reaches an icy patch on which her sis move freely with negligible friction. The difference on altitude between the top of the icy patch and its bottom is 10m. What is the speed of the skier at the bottom of the icy patch. Do you have to know the mass?
To find the speed of the skier at the bottom of the icy patch, we can use the principle of conservation of energy. Since friction is negligible on the icy patch, the only forces acting on the skier are gravity and the skier's initial velocity. We don't need to know the mass of the skier because it cancels out in the calculation.
The principle of conservation of energy states that the total mechanical energy of the skier remains constant throughout the motion. The mechanical energy can be in the form of kinetic energy (KE) or potential energy (PE).
Initially, when the skier is sliding downhill, she has gravitational potential energy (GPE) due to her height above the icy patch. At the bottom of the icy patch, all the GPE is converted into kinetic energy. Therefore, we can equate the initial GPE to the final kinetic energy (KE) to find the final velocity.
The formula to calculate gravitational potential energy is given by:
GPE = mgh
where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height difference between the top and bottom of the icy patch (10 m).
Now, the formula for kinetic energy is given by:
KE = (1/2)mv²
where v is the final velocity of the skier at the bottom of the icy patch.
Equating the initial GPE to the final KE:
mgh = (1/2)mv²
The mass (m) is present on both sides of the equation and can be canceled out:
gh = (1/2)v²
We can rearrange this equation to find v:
v² = 2gh
Taking the square root of both sides:
v = √(2gh)
Substituting the values:
v = √(2 * 9.8 m/s² * 10 m)
v = √(196 m²/s²)
v = 14 m/s
Therefore, the speed of the skier at the bottom of the icy patch is 14 m/s, independent of the skier's mass.