A man skis down a slope 120 m high. If 80 percent of his initial potential energy is lost to friction and air resistence, what is his speed at the bottom of the slope?
20% of M g H is converted to (1/2) M V^2
Note that the mass M cancels out.
0.2 gH = 0.5 V^2
H = 120 m; g = 9.81 m/s^2
Now just Solve for V
45
PE=KE
mgh=1/2mv^2
0.2(9.8m/s^2)(120m)=0.5(v^2)
(235.2m^2/s^2)/0.5=(0.5v^2)/0.5
v^2=470.4
v=21.689 m/s
To determine the speed of the man at the bottom of the slope, you can make use of the principle of conservation of mechanical energy. According to this principle, the total mechanical energy, which is the sum of potential energy and kinetic energy, remains constant in the absence of external forces like friction.
Given:
- Height of the slope (change in potential energy), h = 120 m
- Loss of potential energy due to friction and air resistance = 80% = 0.8 (as a decimal)
First, calculate the loss in potential energy using the formula:
Loss in Potential Energy = Loss Factor * Initial Potential Energy
The initial potential energy is equal to the change in potential energy at the top of the slope:
Initial Potential Energy = m * g * h
where:
m = mass of the skier (which we can disregard since it cancels out later in the calculations)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the given values, the initial potential energy becomes:
Initial Potential Energy = 9.8 m/s^2 * 120 m = 1176 J (Joules)
Now, we can calculate the loss in potential energy:
Loss in Potential Energy = 0.8 * 1176 J = 940.8 J
As per the principle of conservation of mechanical energy, this loss will be equal to the gain in kinetic energy:
Loss in Potential Energy = Gain in Kinetic Energy
Kinetic Energy at the bottom of the slope can be calculated using the formula:
Kinetic Energy = 1/2 * m * v^2
where:
v = velocity (speed) of the skier
Rearranging the equation to solve for v:
v^2 = 2 * (Loss in Potential Energy) / m
Since mass (m) cancels out, we can ignore it. Thus, the equation becomes:
v^2 = 2 * Loss in Potential Energy
Now, substituting the value of the loss in potential energy:
v^2 = 2 * 940.8 J
v^2 = 1881.6 J
Finally, take the square root of both sides to get the velocity:
v = √(1881.6 J) ≈ 43.4 m/s
So, the speed of the man at the bottom of the slope is approximately 43.4 m/s.