Suppose a scrawny 20. kg Wyle was shot straight up with an initial velocity of +50 m/s.
a. Assuming that all his initial Ek was transformed into Eg, what is the maximum height he could reach?
b. Suppose that 20% of his initial Ek were lost due to friction with the air (air resistance).
What is the maximum height he could reach?
1/2 mv^2 = mgh
or, v^2 = 2gh
Would that be 127.6 m and 102.08 m
To determine the maximum height Wyle could reach in both scenarios, we need to analyze the energy transformations and conservation of energy in his motion.
a. Assuming all his initial kinetic energy (Ek) is transformed into gravitational potential energy (Eg), we can calculate the maximum height using the conservation of energy principle. The equation for gravitational potential energy is:
Eg = m * g * h
Where:
m = mass of Wyle (20 kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = maximum height
Since all the initial kinetic energy is transformed into potential energy, we can write:
Ek = Eg
So, the equation becomes:
(1/2) * m * v^2 = m * g * h
Plugging in the given values:
(1/2) * 20 kg * (50 m/s)^2 = 20 kg * 9.8 m/s^2 * h
Simplifying the equation:
(1/2) * 2500 kg m^2/s^2 = 196 kg m/s^2 * h
Now, we can solve for h:
1250 kg m^2/s^2 = 196 kg m/s^2 * h
h = (1250 kg m^2/s^2) / (196 kg m/s^2)
h ≈ 6.377 m
Therefore, the maximum height Wyle could reach, assuming all initial kinetic energy is transformed into gravitational potential energy, is approximately 6.377 meters.
b. If 20% of his initial kinetic energy is lost due to friction with the air, we need to account for this loss when calculating the maximum height. The remaining kinetic energy after the loss can be calculated as:
Remaining Ek = Initial Ek * (1 - 20/100)
Plugging in the values:
Remaining Ek = (1/2) * 20 kg * (50 m/s)^2 * (1 - 0.20)
Simplifying the equation:
Remaining Ek = (1/2) * 20 kg * (50 m/s)^2 * 0.8
The remaining kinetic energy is the same as the initial kinetic energy transformed into gravitational potential energy.
So, the equation becomes:
(1/2) * m * v^2 * 0.8 = m * g * h
Plugging in the given values:
(1/2) * 20 kg * (50 m/s)^2 * 0.8 = 20 kg * 9.8 m/s^2 * h
Simplifying the equation:
(1/2) * 2500 kg m^2/s^2 * 0.8 = 196 kg m/s^2 * h
Now, we can solve for h:
1000 kg m^2/s^2 = 196 kg m/s^2 * h
h = (1000 kg m^2/s^2) / (196 kg m/s^2)
h ≈ 5.102 m
Therefore, the maximum height Wyle could reach, considering the 20% loss of initial kinetic energy due to friction with the air, is approximately 5.102 meters.