This problem is about escape velocity, which is the minimum velocity required for an object to escape the gravitational pull of a planet (in this case, Earth) and never return. Here's how to calculate it:

Understanding Escape Velocity

* Gravitational Potential Energy: As an object moves further away from Earth, its gravitational potential energy increases.

* Kinetic Energy: To escape Earth's gravity, the object needs enough kinetic energy to overcome the gravitational potential energy.

* Balance: At escape velocity, the object's kinetic energy is equal to the gravitational potential energy at infinity (where the gravitational force is considered zero).

Derivation

1. Gravitational Potential Energy at Infinity:

- The potential energy at infinity is defined as zero.

- The change in potential energy from the surface of Earth to infinity is given by:

- ΔPE = GMm/R

Where:

- G is the gravitational constant (6.674 x 10^-11 m^3 kg^-1 s^-2)

- M is the mass of Earth (5.972 x 10^24 kg)

- m is the mass of the object

- R is the radius of Earth (6.371 x 10^6 m)

2. Kinetic Energy at Escape Velocity:

- KE = (1/2)mv^2

- v is the escape velocity

3. Equating Kinetic and Potential Energy:

- (1/2)mv^2 = GMm/R

4. Solving for Escape Velocity:

- v^2 = 2GM/R

- v = √(2GM/R)

Escape Velocity from Height 'h'

If the object is launched from a height 'h' above the surface of Earth, the effective distance from the center of Earth becomes R + h. Therefore, the escape velocity from height 'h' is:

v = √(2GM/(R + h))

Important Notes

* This formula assumes no air resistance. In reality, air resistance will significantly affect the required velocity.

* The escape velocity does not depend on the mass of the object. This is because the gravitational force and the required kinetic energy both scale proportionally with the object's mass.

Let me know if you'd like a numerical example or want to explore further concepts related to escape velocity!