Understanding the Concepts
* Escape Velocity: The minimum speed an object needs to escape the gravitational pull of a planet or other celestial body and never return.
* Gravitational Potential Energy: The energy an object possesses due to its position in a gravitational field.
* Kinetic Energy: The energy an object possesses due to its motion.
The Calculation
1. Conservation of Energy: The key is to use the principle of conservation of energy. As the projectile moves away from Earth, its gravitational potential energy increases, while its kinetic energy decreases. At escape velocity, the projectile's kinetic energy will be zero infinitely far away from Earth.
2. Setting up the Equation:
* Initial Kinetic Energy (KE) + Initial Gravitational Potential Energy (GPE) = Final KE + Final GPE
* (1/2)mv² - GMm/R = 0 + 0
Where:
* m = mass of the projectile
* v = escape velocity
* G = gravitational constant (6.674 × 10⁻¹¹ m³/kg s²)
* M = mass of Earth (5.972 × 10²⁴ kg)
* R = radius of Earth (6.371 × 10⁶ m)
3. Solving for Escape Velocity:
* (1/2)mv² = GMm/R
* v² = 2GM/R
* v = √(2GM/R)
4. Plugging in the Values:
* v = √(2 * 6.674 × 10⁻¹¹ m³/kg s² * 5.972 × 10²⁴ kg / 6.371 × 10⁶ m)
* v ≈ 11,180 m/s
Therefore, the minimum initial speed a projectile must have at Earth's surface to escape gravitational pull (ignoring air resistance) is approximately 11,180 m/s (or about 25,000 mph).
