WHAT IS ESCAPE VELOCITY?

Saket KUMAR 

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WHAT IS ESCAPE VELOCITY ?

Escape velocity is a minimum velocity with which a body should be projected so that it overcomes the gravitational pull of the earth. Escape velocity is defined to be the minimum velocity an object must have in order to escape the gravitational field of the earth, that is, escape the earth without ever falling back. An object which has this velocity at the surface of the earth, will totally escape the earth's gravitational field ignoring the losses due to the atmosphere. Escape velocity is the speed that an object needs to be traveling to break free of a planet or moon's gravity well and leave it without further propulsion. 

For example, a spacecraft leaving the surface of Earth needs to be going 7 miles per second, or nearly 25,000 miles per hour to leave without falling back to the surface or falling into orbit.

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🔹How to Calculate Escape Velocity
Author Info

The escape velocity is the velocity necessary for an object to overcome the gravitational pull of the planet that object is on.[1] For example, a rocket going into space needs to reach the escape velocity in order to make it off Earth and get into space.

•Part One of Two:
Understanding Escape Velocity

1
Define escape velocity. Escape velocity is the velocity of an object required to overcome the gravitational pull of the planet that object is on to escape into space.[2] A larger planet has more mass and requires a much greater escape velocity than a smaller planet with less mass.[3]
 2
Begin with conservation of energy. Conservation of energy states that the total energy of an isolated system remains unchanged. In the derivation below, we will work with an Earth-rocket system and assume that this system is isolated.
In conservation of energy, we equate the initial and final potential and kinetic energies K_{{1}}+U_{{1}}=K_{{2}}+U_{{2}}, where K is kinetic energy and U is potential energy.
3
Define kinetic and potential energy.
Kinetic energy is energy of motion, and is equal to {\frac {1}{2}}mv^{{2}}, where m is the mass of the rocket and v is its velocity.
Potential energy is energy that results from where an object is relative to the bodies in the system. In physics, we typically define the potential energy to be 0 at an infinite distance from Earth. Since the gravitational force is attractive, the potential energy of the rocket will always be negative (and smaller the closer it is to Earth). Potential energy in the Earth-rocket system is thus written as -{\frac {GMm}{r}}, where G is Newton's gravitational constant, M is the mass of Earth, and r is the distance between the two masses' centers.
4
Substitute these expressions into conservation of energy. When the rocket achieves the minimum velocity required to escape Earth, it will eventually stop at an infinite distance from Earth, so K_{{2}}=0. Then, the rocket will not feel Earth's gravitational pull and will never fall back to Earth, so U_{{2}}=0 as well.
{\frac {1}{2}}mv^{{2}}-{\frac {GMm}{r}}=0
5
Solve for v.
{\begin{aligned}{\frac {1}{2}}mv^{{2}}&={\frac {GMm}{r}}\\v^{{2}}&={\frac {2GM}{r}}\\v&={\sqrt {{\frac {2GM}{r}}}}\end{aligned}}
v in the above equation is the escape velocity of the rocket - the minimum velocity required to escape the gravitational pull of Earth.
Note that the escape velocity is independent of the mass of the rocket m. The mass is reflected in both the potential energy provided by Earth's gravity as well as the kinetic energy provided by the movement of the rocket.
•Part Two of Two:
Calculating Escape Velocity
1
State the equation for escape velocity.
v={\sqrt {{\frac {2GM}{r}}}}
The equation assumes the planet you are on is spherical and has constant density.[4] In the real world, the escape velocity depends on where you are at on the surface because a planet bulges at the equator due to its rotation and has slightly varying density due to its composition.
2
Understand the variables of the equation.
G=6.67\times 10^{{-11}}{{\rm {\ N\ m^{{2}}\ kg^{{-2}}}}} is Newton’s gravitational constant. The value of this constant reflects the fact that gravity is an incredibly weak force. It was determined experimentally by Henry Cavendish in 1798,[5] but has proven to be notoriously difficult to measure precisely.
G can be written using only base units as 6.67\times 10^{{-11}}{{\rm {\ m^{3}\ kg^{{-1}}\ s^{{-2}}}}}, since 1{{\rm {\ N}}}=1{{\rm {\ kg\ m\ s^{{-2}}}}}.[6]
Mass M and radius r are dependent upon the planet you wish to escape from.
You must convert to SI units. That is, mass is in kilograms (kg) and distance is in meters (m). If you find values that are in different units, such as miles, convert them to SI.
3
Determine the mass and radius of the planet you are on. For Earth, assuming that you are at sea level, r=6.38\times 10^{{6}}{{\rm {\ m}}} and M=5.98\times 10^{{24}}{{\rm {\ kg}}}.[7]
Search online for a table of masses and radii for other planets or moons.
4
Substitute values into the equation. Now that you have the necessary information, you can start solving the equation.
v={\sqrt {{\frac {2(6.67\times 10^{{-11}}{{\rm {\ m^{{3}}\ kg^{{-1}}\ s^{{-2}}}}})(5.98\times 10^{{24}}{{\rm {\ kg}}})}{(6.38\times 10^{{6}}{{\rm {\ m}}})}}}}
5
Evaluate. Remember to evaluate your units at the same time and cancel them out as needed to obtain a dimensionally consistent solution.
{\begin{aligned}v&={\sqrt {{\frac {2(6.67)(5.98)}{(6.38)}}\times 10^{{7}}{{\rm {\ m^{{2}}\ s^{{-2}}}}}}}\\&\approx 11200{{\rm {\ m\ s^{{-1}}}}}\\&=11.2{{\rm {\ km\ s^{{-1}}}}}\end{aligned}}
In the last step, we converted the answer from SI units to {{\rm {\ km\ s^{{-1}}}}} by multiplying by the conversion factor {\frac {{\text{1 km}}}{{\text{1000 m}}}}. ✅✅

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ESCAPE VELOCITY FORMULA:-

   Escape velocity formula is given by:-

Where,
Vesc is the escape velocity,
G is the gravitational constant is 6.673 × 10-11m3kg-1s-2,
M is the mass of the planet,
R is the radius from center of gravity.

Acceleration due to gravity formula is
Where, g is acceleration due to gravity of earth.

Hence Escape velocity is also given as
It is expressed in m/s and escape velocity of earth is 11,200 m/s.

Escape velocity formula is helpful in finding escape velocity of any body or planet, if mass and radius is known. It has wide applications in space calculations.

                      Or

Escape Velocity Problems

Problems based on escape velocity are given below :

Solved Examples

Question 1: Calculate the escape velocity of the Jupitor if its mass is 1.89813 ×$\times$ 1027 Kg and radius is 71492 Km.
Solution:

Given: Mass M = 1.89813 ×$\times$ 1027Kg,
          Radius R = 71492 Km
          Gravitational Constant G = 6.673 ×$\times$ 10-11m3kg-1s-2
Hence Escape Velocity is given by
Vesc = 2GMR−−−−√$\sqrt{\frac{2GM}{R}}$
        = $\sqrt{\frac{2 \times 6.673 \times 10^{-11} \times 1.898 \times 10^{27} }{(71492)}}$
        = 59.5 Km/s.

Question 2: Calculate the escape velocity of the moon if Mass is 7.35 ×$\times$1022 Kg and radius is 1.7 ×$\times$ 106 m.
Solution:

M = 7.35 ×$\times$ 1022 Kg,
R = 1.7 ×$\times$ 106 m
Hence Escape Velocity is given by Vesc = 2GMR−−−−√$\sqrt{\frac{2GM}{R}}$
                                                           = 2×6.673×10−11×7.35×10221.7×106−−−−−−−−−−−−−−−−√$\sqrt{\frac{2\times 6.673\times {10}^{-11}\times 7.35\times {10}^{22}}{1.7\times {10}^6}}$
                                                           = 2.4 ×$\times$ 103 m/s.

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