Mt. Everest is more than 8,000 m high. How fast would an object be moving if it could free fall to sea level after being releases from an 8,000-m elevation? (Ignore air resistance)
To find the speed at which an object would be moving if it were to free fall from an 8,000-meter elevation, we can use the principle of conservation of energy. At the top of the mountain, the object would have only potential energy (PE) and no kinetic energy (KE). As it falls to sea level, its potential energy is converted into kinetic energy.
The potential energy of an object can be calculated using the formula: PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
In this case, the height (h) is 8,000 meters. The potential energy at the top of the mountain is equal to the kinetic energy at sea level, so we can equate the two:
PE = KE
mgh = 0.5mv²
Canceling out the mass (m) on both sides:
gh = 0.5v²
Simplifying the equation:
v² = 2gh
Now we can solve for v:
v = √(2gh)
v = √(2 * 9.8 m/s² * 8,000 m)
v ≈ √(15.4 * 8,000)
v ≈ √123,200
v ≈ 351.27 m/s
Therefore, if an object were to free fall from an 8,000-meter elevation, it would be moving at a speed of approximately 351.27 meters per second.
To determine the speed of an object free-falling from a certain height, we can use the concept of potential energy and kinetic energy.
When the object is at the top of Mt. Everest (8,000 m elevation), it has potential energy due to its height above the ground. As the object falls, this potential energy is converted into kinetic energy, which is the energy of motion. At the bottom of the fall (sea level), all the potential energy is converted into kinetic energy.
The formula to calculate the potential energy at a certain height is given by:
Potential Energy = mass × acceleration due to gravity × height
Given that the acceleration due to gravity is approximately 9.8 m/s², and the height is 8,000 m, the potential energy at the top of Mt. Everest would be:
Potential Energy = mass × 9.8 m/s² × 8,000 m
To find the speed of the object when it reaches sea level, we equate this potential energy to the kinetic energy using the equation:
Potential Energy = Kinetic Energy
Potential Energy = ((1/2) × mass × velocity²)
Since we are only interested in the speed, we can set the two equations equal to each other:
mass × 9.8 m/s² × 8,000 m = (1/2) × mass × velocity²
By canceling out the mass on both sides, we get:
9.8 m/s² × 8,000 m = (1/2) × velocity²
Simplifying further:
(9.8 m/s² × 8,000 m) / (1/2) = velocity²
velocity² = (9.8 m/s² × 8,000 m) / (1/2)
Now we can calculate the speed by taking the square root of the right side of the equation:
velocity = √((9.8 m/s² × 8,000 m) / (1/2))
Evaluating the expression:
velocity = √(78,400 m²/s² / (1/2))
velocity = √(78,400 m²/s² × 2)
velocity = √(156,800 m²/s²)
velocity ≈ 396.28 m/s
Therefore, an object released from an 8,000 m elevation (Mt. Everest) and free-falling to sea level would be moving at a speed of approximately 396.28 meters per second.