Explain:Escape velocity and derive mathematically
Answers
Answer:
Definition of escape velocity. : the minimum velocity that a moving body (such as a rocket) must have to escape from the gravitational field of a celestial body (such as the earth) and move outward into space.
Explanation:
The formula for escape velocity comprises of a constant, G, which we refer to as the universal gravitational constant. The value of it is = 6.673 × 10-11 N . m2 / kg2. The unit for escape velocity is meters per second (m/s).
Explanation
Escape Velocity, it's Formula and Derivation
When an object is thrown vertically upwards, it reaches a certain height and comes back to the earth. But when it is given greater initial velocity, it reaches greater height before coming back.
An object can be thrown up with a certain minimum initial velocity so that, the object goes beyond the earth's gravitational field and escape from earth, this velocity is known as escape velocity of the earth. So, escape velocity is defined as the minimum initial velocity that will take a body away above the surface of a planet when it's projected vertically upwards.
On throwing the object upwards, work has to be done against the gravity. If a certain minimum velocity is given to an object, such that the work done against gravity from the planet's surface to infinity(outside gravitational field of planet) is equal to the kinetic energy of the object, then it will not return back to the planet.
Derivation of Escape Velocity
Suppose that the planet be a perfect sphere of radius R having mass M. Let a body of mass m is to be projected from point A on the earth's surface as shown in the figure. Join OA and produce it further. Let us take two points P and Q which are at distances x and (x + dx) from the center of the earth.
To calculate the escape velocity of the earth, let the minimum velocity to escape from the earth's surface be ve. Then, kinetic energy of the object of mass m is
When the projected object is at point P which is at a distance x from the center of the earth, the force of gravity between the object and earth is
Work done in taking the body against gravitational attraction from P to Q is given by