Escape Velocity: Definition, Formula, escape velocity list

INTRODUCTION OF ESCAPE VELOCITY

If an object is thrown straight up from the earth’s surface, the object rises to a certain height and falls to the earth’s surface due to gravity. At maximum height the velocity of the object is zero. The harder the object is thrown i.e. the higher the projectile velocity, the higher the velocity of the object after rising to zero. As the projectile velocity continues to increase, a state is reached where the velocity of the object is no longer zero (for a finite height), i.e. it no longer returns to the surface. In this case, the object moves further away from the Earth and the gravitational attraction force of the Earth on the object becomes less and less. So, we can say, the object reaches infinite distance from Earth. The minimum projectile velocity required for this to occur is the release velocity. Notably, the concept of velocity applies not only to objects launched from Earth, but also to objects launched from the surface of any planet or satellite.

ESCAPE VELOCITY or ESCAPE SPEED

DEFINATION: The minimum velocity at which an object can be thrown from the surface of the Earth or any other planet or satellite beyond its gravitational attraction is called the escape velocity or escape speed of the object in the case of that planet or satellite.
CALCULATION: An object falling outside the gravitational pull of a planet or satellite means that the object no longer returns to the surface of the planet or satellite. Let us assume that Earth’s mass = M and Earth’s radius = R. Gravitational force of attraction on an object of mass m at a distance x from the centre of the earth = (GMm/X²) dx.

Therefore, the amount of work done to send the object an infinite distance from the surface is

If the release velocity of the object from Earth’s surface is v_e, then the initial kinetic energy of the object = 1/2mv_e². When this kinetic energy is equal to the above function (GMn/R), the object escapes the gravitational attraction of the Earth.

So, 1/2mv_e² = GMn/R or, v_e² = 2GM/R

or, v_e = √(2GM/R) ……….(I)

Gravitational acceleration at the surface,

g = GM/R² or, GM = gR²

So, v_e = √2gR

This is the value of the escape velocity of any object from the Earth’s surface. Note that this escape velocity does not depend on the mass m of the object. That is, the value of the escape velocity is the same for all light and heavy objects.

We know that the gravitational acceleration at the surface is g = 9.8 m/s² and the radius of the earth is R = 6400 km = 6.4 X 10⁶ m.

So, v_e = √ (2 X 9.8 X 6.4 X 10⁶) = 11200 m/s = 11.2 km/s.

So, if an object of any mass is thrown upwards from the Earth’s surface with a velocity of 11.2 km/s, the object will leave the Earth’s gravitational field, i.e. it will not return to the Earth’s surface.

CHART OF ESCAPE VELOCITY OF EVERY MEMBER OF OUR SOLAR SYSTEM

Sl. No.	Name of Members	Gravitational Acceleration (m/s²)	Radius (m)	Escape Velocity Ve (m/s)
1.	Sun	274	6.96 X 10^9	617.6
2.	Mercury	3.7	2.4 X 10^6	4.25
3.	Venus	8.87	6.05 X 10^6	10.36
4.	Earth	9.807	6.37 X 10^6	11.2
5.	Mars	3.71	3.38 X 10^6	5.03
6.	Jupiter	24.79	7 X 10^7	59.5
7.	Saturn	10.44	5.82 X 10^7	35.5
8.	Uranus	8.87	2.53 X 10^7	21.3
9.	Neptune	11.15	2.46 X 10^7	23.5

SCARCITY OF LIGHT GASES IN THE EARTH'S ATMOSPHERE

At the time of Earth’s creation, its atmosphere contained heavy gases such as oxygen(O), nitrogen(N), and lighter gases such as hydrogen(H) and helium(He). But now hydrogen(H), helium(He) etc. gases do not exist in the atmosphere. This phenomenon can be explained from the release velocity.

The kinetic theory of gases shows that the root mean square velocity (rms velocity) of hydrogen(H) gas molecules at a ‘Standard Temperature and Pressure’ (at STP) is about 1.6 km/s. At the time of Earth’s creation, the Earth’s surface was much warmer, so the rms velocity of hydrogen(H) molecules at that time was close to 5 km/s. This means that some hydrogen(H) molecules had velocities equal to or greater than the escape velocity of 11 km/s and some molecules had lower velocities. As a result, over a long period of time, most of the hydrogen(H) molecules have moved out of the Earth’s gravitational field, i.e., leaving the Earth and disappearing into space. The same thing happened with helium(He) gas. On the other hand, the rms velocity of heavy gas molecules like oxygen(O), nitrogen(N), carbon dioxide(CO2) etc. is very low and since this velocity value is less than the escape speed, it is not possible for them to move out of the gravitational field of the earth. As a result, the earth’s atmosphere is formed by them.

As the ratio of mass and radius of Mercury or Moon is relatively low, the value of release velocity is relatively low in their case. Therefore, since the velocity of the gas molecules, light or heavy, is greater than the release velocity, they are able to move out of their field of attraction. That is why neither Mercury nor the Moon have an atmosphere. In the case of planets like Jupiter, Saturn, etc., the situation is just the opposite, i.e. because the escape velocity from these planets is very high, no light or heavy gas can escape out of their gravitational field. Hence, the atmospheres of these planets contain large amounts of hydrogen and helium gas.

CONCLUSION

Escape velocity is the lowest speed at which a body needs move in order to break free of the earth’s gravitational pull.
Let’s say we toss a ball and it bounces back. This is because the ball is pulled toward the earth’s surface due to the gravitational force the earth’s surface exerts on it.
Our objective is accomplished if we raise the velocity to the point where the thrown object never comes back.
Escape velocity is the term used to describe this speed.
The ball is thrown skyward, yet gravity pulls it back down to the earth.
The same ball is launched fast enough to avoid returning to Earth due to gravity. Escape velocity is the term used to describe this speed.

The necessity of Escape Velocity: Since gravity is the sole factor affecting you, your escape velocity is calculated after the rocket engines—or whatever propelled you to travel at 11 km/s—have stopped. Imagine yourself leaving the earth at a high speed, far above the atmosphere. You are slowed down by the Earth’s gravitational pull, but it becomes less strong the farther you are from it. Earth’s gravitational pull would eventually cause you to stop if your speed exceeded the escape velocity, at which point you would fall back to the planet. The speed at which you can escape your fate and carry on traveling

REMINDER ON THIS TOPIC

The force by which any two particles of matter in this universe attract each other is called gravitation.
Any two particles in the universe attract each other along their connecting straight lines. The value of this attractive force is proportional to the product of the masses of the two particles and inversely proportional to the square of the distance.
The force with which two particles of unit mass attract each other at a distance of unit distance is called the gravitational constant.
The gravitational force acting on an object of unit mass at any point in a gravitational field is called the gravitational field strength at that point.
The amount of work done by an external agent to bring an object of unit mass from an infinite distance to a point in the gravitational field is called the gravitational field at that point.
The force with which the earth attracts any object on or near the surface is called gravity. The acceleration due to the force of gravity on a freely falling object is called gravitational acceleration.
Kepler’s Laws of Motion of Planets and Satellites:

First Law: Each planet revolves around the Sun in an elliptical orbit and the Sun is at one focus of that ellipse.
Second Law: A straight line joining the Sun and a planet traverses equal areas in equal periods of time.
Third Law: The square of the orbital period of a planet around the Sun is proportional to the cube of the semi-radius of the orbit.

The minimum velocity at which an object can be thrown from the surface of the Earth or any other planet or satellite beyond its gravitational attraction is called the release velocity.
If the relative angular velocity of an artificial satellite with respect to the Earth’s angular momentum is zero and the satellite is always at the equator, it appears from the surface that the satellite is fixed in the same place in the sky. Such satellites are called geosynchronous satellites.