By landing properly and on soft ground (and by being lucky!), humans have survived falls from airplanes when, for example, a parachute failed to open, with astonishingly little injury. Without a parachute, a typical human eventually reaches a terminal velocity of about 62.0 m/s. Suppose the fall is from an airplane 1000 m high.
How fast would a person be falling when he reached the ground if there were no air drag?
(in m/s)
If a 70.0 kg person reaches the ground traveling at the terminal velocity of 62.0 m/s, how much mechanical energy was lost during the fall?
(1/2) m v^2 = m g h
so
v = sqrt( 2 g h ) = sqrt (2 * 9.81 * 1000 ) = 140 m/s
Potential energy at top = m g h = 70 * 9.81 *1000 = 686,700 Joules
Kinetic energy at ground = (1/2) m v^2 = 35 * 62^2 = 134,540 Joules
lost to friction= 552,160 Joules warming air
To calculate the speed at which a person would fall when reaching the ground without air drag, we can make use of the laws of motion.
The initial height (h) from the airplane is given as 1000 m, and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
Using the equation of motion:
v^2 = u^2 + 2gh
Where:
v = final velocity (unknown)
u = initial velocity (0 m/s, as the person is initially stationary)
g = acceleration due to gravity (9.8 m/s^2)
h = height (1000 m)
Rearranging the equation:
v^2 = 0 + 2(9.8)(1000)
v^2 = 19600
Taking the square root of both sides:
v ≈ √(19600)
v ≈ 140 m/s
Therefore, if there were no air drag, a person would be falling at approximately 140 m/s when they reached the ground.
Next, to calculate the amount of mechanical energy lost during the fall, we can use the formula for potential energy:
PE = mgh
Where:
PE = potential energy (unknown)
m = mass of the person (70.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height (1000 m)
Substituting the given values into the formula:
PE = (70.0)(9.8)(1000)
PE ≈ 686,000 J
Therefore, approximately 686,000 Joules of mechanical energy would be lost during the fall.
To calculate the speed at which a person would be falling when they reach the ground without air drag, we can use the principles of physics.
1. The initial velocity of the person is 0 m/s as they are falling from rest.
2. The final velocity can be calculated using the formula v = √(2gh), where v is the final velocity, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which they fell.
3. Plugging in the values, we have v = √(2 * 9.8 m/s² * 1000 m).
4. Calculate the final velocity using a calculator or by hand: v = 44.3 m/s.
So, if there were no air drag, a person would be falling at a speed of 44.3 m/s when they reach the ground.
Next, let's calculate the amount of mechanical energy lost during the fall. We can use the principle of conservation of energy, assuming there is no air drag.
1. The initial mechanical energy is equal to the potential energy at the height of 1000 m, given by mgh, where m is the mass of the person and h is the height.
2. The final mechanical energy can be calculated using the kinetic energy at the terminal velocity, given by (1/2)mv², where m is the mass of the person and v is the terminal velocity.
3. Calculate the initial mechanical energy: E_initial = m * g * h = 70.0 kg * 9.8 m/s² * 1000 m.
4. Calculate the final mechanical energy: E_final = (1/2) * 70.0 kg * (62.0 m/s)².
5. Calculate the mechanical energy lost: E_lost = E_initial - E_final.
Using a calculator or by hand, calculate the mechanical energy lost during the fall:
E_lost = (70.0 kg * 9.8 m/s² * 1000 m) - (1/2 * 70.0 kg * (62.0 m/s)²).
The result for the mechanical energy lost will be in joules (J).