Saturday, July 17, 2021

What is Escape Velocity, Its Equation and many more..

When you throw a ball into the sky why does it fly, why does it didn’t go to space, why did it come back to you.

Yes, you are right, because of the earth’s gravity. But if this is because of earth’s gravity, then how do the rockets and other space crafts go in space.

Yes. Again you are right because the rockets are so powerful and also they produce powerful thrusts to go into space.


But if you get in deep there also other things which help the rockets to fly and leave the earth’s gravitational field.

The most important among all is the escape velocity. However, the rockets don’t go according to the earth’s escape velocity.

Because the earth has a thick atmosphere, which causes high air resistance, you might have already learned about the air resistance or friction due to air.



But still, let’s take a short recap

Air resistance is the force/friction caused by small dust particles in the air. The air resistance depends on the area of the body which moving in the air and the motion of that body.

e.g. when you go fast on a bike, you feel a backward force acting on you, right.





e.g. when a skydiver jumps from an aircraft and then he suddenly opens his parachute, he immediately starts flying in the air or he gets slower after opening the parachute. Because the parachute has a larger area.


Now let’s come back to the point

Escape velocity is the minimum velocity that a body must need to leave the earth’s or any other planet’s gravitational field. Every planet has its own and different escape velocities.

The symbol of escape velocity is Ve (Velocity(v) with escape(e)), and the SI unit is km/s.

The escape velocity depends on the mass and the size of the body/planet that we want to escape.


Escape Velocity’s Equation Derivation

We all what is work done, so if we want to escape any planet’s gravitational field, we have to do some work.

So equation to find the work done is:



And as we all the to escape any planet we will also need kinetic energy And the equation to find the kinetic energy is:



But because we are working escape velocity so our velocity in this equations going to be:



And to derive the escape velocity’s equation we can say that the



This means the work we need to done and the kinetic we need to give to the object (to leave the planet’s gravitational field) are equal to each other.

Therefor



Then our escape velocity is going to be:



this is the equation to find the escape velocity, you know how it feels like when you solve any problem, it feels like we are Einstein.

But there is one more way to calculate the escape velocity, which is easier than we have seen above.

Now consider there is an object on earth or any planet, then the gravitational force on that object will be:



Where g is the acceleration due to gravity G is the value of gravity (6.67430 × 10-11). M is the mass of the planet, and R is the radius of the planet.

Now if we merge this equation to the equation that we derived before, the equation we get will be:



Because we have merged the equation to find the acceleration due to gravity to the equation to find the escape velocity. So we can write this same equation in this way:



Where g is the acceleration due to gravity, R is the radius. Now I’m feeling like Issac newton.


How to calculate the escape velocity

Just for example let’s calculate the escape velocity of earth with the second equation just putting some values.

The radius of the earth is 6,371, but just for easiness let’s take it as 6,400. The acceleration due to gravity is 9.8. Now let’s just put these values in the equation.



Because the acceleration due to gravity is in meter/second, but the radius of the earth was in kilometers so we just multiplied the radius by 1000 to convert it into the meters. Because there are 1000 meters in one kilometer.

And then after getting 11,200, we divided it by 1000 so the answer we get is 11.2km/s.

The escape velocity of the earth is 11.2km/s.

Like this, we can also find the escape velocity of other planets such as Mars, Jupiter, and Pluto. There are two to calculate the escape velocity, you can choose whichever you want.

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