What is the Escape Velocity Equation?
The Earth’s gravity pulls everything in and around it towards its core. So any object that goes to space leaving the surface of the Earth, has to travel at a certain velocity to overcome this pull. The velocity needed for an object to leave any spherical body such as the Earth overcoming its gravitational pull is called Escape Velocity. Learn how to calculate the escape velocity using the escape velocity equation.
Calculating Escape Velocity with the Escape Velocity Formula
In order to calculate the Escape velocity needed in a planet, we use this equation of escape velocity-
Ve =√(2GM/R)
Here,
Ve = Escape Velocity
G = Gravitational Constant as derived by Newton.
M = Mass of the concerned object.
R = Distance between the center of gravity, i.e., the center of the planet and the object that wants to escape.
Escape Speed Equation
As you know, velocity does not speed. Speed has no direction. Therefore we also have the formula to calculate the speed needed for an object to escape the planet and go to an infinitely far away place. The Escape Speed Equation is-
Se = √(2GM/R)
Escape Velocity of Earth Formula
Now that you know all about escape velocity formula derivation, let’s talk about the escape velocity of Earth formula-
The Escape Velocity of Earth Formula is, Ve =√(2GM/R)
The Mass of Earth is- 5.9723 × 1024 kg
The distance between the object and the Earth’s gravitational core- 6,371 km
The gravitational constant is G = 6.673×10-11 N m2 kg-2
Now substituting these values in the formula- Ve =√(2GM/R), we will get the escape velocity of Earth and that is- 11.2 KM/s
Escape Energy Formula
Using the escape velocity formula derivation, we can derive the escape energy formula. Escape energy is the kinetic energy needed for a spaceship to escape the Earth. The escape energy formula is-
KE = ½MV²
Here, K.E = Kinetic Energy
M = Weight of The Object
V = Escape Velocity
Escape Velocity Dimensional Formula
The escape velocity dimensional formula is-
M0 L1 T-1
Calculating the escape velocity and escape energy is crucial to launching any spaceship into space. It is only after a spaceship reaches the escape velocity of the earth that it can escape the gravitational force of the Earth.
FAQs on Escape Velocity Formula
1. What exactly is escape velocity in the context of Physics?
Escape velocity is the minimum speed an object must have to completely break free from the gravitational influence of a celestial body, like a planet or a star, without any further propulsion. Essentially, it's the speed required to travel from the surface of the body to an infinite distance away, overcoming its gravitational pull. For Earth, this value is approximately 11.2 km/s.
2. What is the standard formula to calculate the escape velocity of a planet?
The escape velocity (Vₑ) is calculated using the formula: Vₑ = √(2GM/R). In this equation:
- G represents the universal gravitational constant (approximately 6.674 × 10⁻¹¹ N m²/kg²).
- M is the mass of the celestial body (e.g., the Earth).
- R is the radius of the celestial body, measured from its centre to the object's starting point on the surface.
3. Is there an alternative formula for escape velocity using acceleration due to gravity (g)?
Yes, there is a useful alternative formula. Since the acceleration due to gravity at the surface of a planet is given by g = GM/R², we can substitute GM = gR² into the primary escape velocity formula. This gives us a simpler expression: Vₑ = √(2gR). This formula directly relates escape velocity to the local acceleration due to gravity and the planet's radius.
4. Does the escape velocity of a rocket depend on its own mass or the angle at which it is launched?
No, surprisingly, escape velocity is independent of the mass of the escaping object and its launch direction. The formula Vₑ = √(2GM/R) only depends on the mass (M) and radius (R) of the celestial body (like Earth). This means a small satellite and a massive spaceship require the same minimum speed to escape Earth's gravity, assuming atmospheric drag is ignored.
5. What is the fundamental difference between escape velocity and orbital velocity?
The key difference lies in their purpose and energy balance:
- Orbital Velocity: This is the speed required for an object to maintain a stable orbit around a celestial body. At this speed, the object continuously 'falls' towards the body but has enough horizontal velocity to miss it, resulting in a circular or elliptical path.
- Escape Velocity: This is a higher speed required to overcome the gravitational pull entirely and leave the orbit, never to return. An object reaching escape velocity has enough kinetic energy to overcome its negative gravitational potential energy, allowing it to reach an infinite distance.
6. On what physical principle is the escape velocity formula derived?
The derivation of the escape velocity formula is based on the principle of conservation of energy. To escape the gravitational field, the object's total mechanical energy (sum of its kinetic and potential energy) must be zero. The initial kinetic energy (½mv²) must be exactly enough to counteract the initial negative gravitational potential energy (-GMm/R) to bring the total energy to zero, allowing the object to reach infinity with zero velocity.
7. How is the escape velocity for Earth calculated to be 11.2 km/s?
We can calculate this using the formula Vₑ = √(2GM/R) and substituting the standard values for Earth:
- G ≈ 6.674 × 10⁻¹¹ N m²/kg²
- M (Mass of Earth) ≈ 5.972 × 10²⁴ kg
- R (Radius of Earth) ≈ 6.371 × 10⁶ m
8. What happens if an object is launched at a speed greater than the escape velocity?
If an object is launched with a speed exceeding the escape velocity, it will not only escape the planet's gravitational field but will also have some leftover kinetic energy when it reaches an 'infinite' distance. This means it will continue moving through space with a residual velocity, rather than just reaching a theoretical stop at infinity.
9. What is the dimensional formula for escape velocity?
Since escape velocity is a measure of speed, its dimensional formula is the same as that for velocity. It does not depend on mass and is defined by length per unit time. Therefore, the dimensional formula for escape velocity is [M⁰ L¹ T⁻¹].
