A skier skis from rest from a vertical height h1 = 19.1 m over two successively lower hills of vertical heights h2 = 15 m and h3 = 5.2 m. (Fig. 7-30). The summit of the third hill fits a circle of radius h3 centered at height 0 m. Friction with the snow and air resistance are negligible.
(b) Does the skier leave the surface at x3?
Yes/No
If not, what should h1 be so that she just leaves the surface at x3? If yes, find the maximum value of h1 at which the skier stays on. (Ignore the middle hill in this calculation.)
_________m
so the skier does leave the surface at X3 but i don't know why.
To determine if the skier leaves the surface at x3, we need to compare the potential energy of the skier at the summit of the third hill (h3) with the kinetic energy required to leave the surface.
At the summit of the third hill, the potential energy of the skier is given by:
PE3 = m * g * h3
where m is the mass of the skier and g is the acceleration due to gravity.
The kinetic energy required to leave the surface is given by:
KE = (1/2) * m * v^2
where v is the velocity of the skier.
If the potential energy at the summit of the third hill (PE3) is greater than or equal to the kinetic energy required to leave the surface (KE), then the skier leaves the surface at x3.
To find the maximum value of h1 at which the skier stays on, we need to ensure that the potential energy at the summit of the third hill (PE3) is less than the kinetic energy required to leave the surface (KE).
In this case, we can assume that the skier just leaves the surface at x3. Therefore, to find the required height of h1, we need to find the kinetic energy at x3 and equate it to the potential energy at the summit of the third hill (PE3):
(1/2) * m * v^2 = m * g * h3
Simplifying the equation, we get:
v^2 = 2 * g * h3
Now we can solve for the velocity (v) at x3 using the equation:
v = sqrt(2 * g * h3)
Substituting the value of v back into the equation for kinetic energy, we get:
KE = (1/2) * m * (2 * g * h3)
Since the skier just leaves the surface at x3, the initial potential energy (PE1) of the skier can be equated to the kinetic energy at x3 (KE).
PE1 = KE = (1/2) * m * (2 * g * h3)
Substituting the values of g and h3, we can solve for h1:
PE1 = (1/2) * m * (2 * (9.8 m/s^2) * 5.2 m)
Simplifying the equation, we get:
PE1 = 50.96 * m
Therefore, the required h1 for the skier to just leave the surface at x3 is 50.96 meters.
To determine whether the skier leaves the surface at x3, we need to compare the potential energy of the skier at the highest point of the third hill (which is equal to her initial potential energy) with the maximum potential energy she can have at that point without leaving the surface.
Let's analyze the situation step by step:
1. Calculate the initial potential energy at the starting point:
Potential energy at starting point = mgh1
where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s²), and h1 is the height of the first hill (19.1 m).
2. Calculate the maximum potential energy at the highest point of the third hill without leaving the surface:
Maximum potential energy = mgh3
where h3 is the height of the third hill (5.2 m).
If the initial potential energy is greater than the maximum potential energy, it means the skier has enough energy to leave the surface. Otherwise, if the initial potential energy is less than or equal to the maximum potential energy, the skier will stay on the surface.
Now, let's substitute the given values and calculate:
Initial potential energy = mgh1 = m * 9.8 m/s² * 19.1 m
Maximum potential energy = mgh3 = m * 9.8 m/s² * 5.2 m
Comparing the two values will give us the answer.