Hello students, here we are going to discuss the energy and velocity that is required by body so as to get escaped from the earth’s gravitational influence. We know that due to gravitational potential energy object moving in upward direction comes back to the earth’s surface. Gravitational potential energy is always directed towards the centre of earth. We need to throw the object at very high speed to remove it from earth’s influence and object will come back to surface of earth. The magnitude of gravitational potential energy at height ‘h’ is given as,
Negative sing indicates that the object is bound to surface of earth.
Definition of escape velocity
The minimum velocity required by body so that it can escape from earth’s gravitational influence is called as escape speed. It is denoted as ve
Derivation of formula for escape speed
Let us find the formula for escape speed of object using law of conservation of energy.
Consider a body of mass m to be projected from the surface of earth.
Let ‘M’ be the mass of earth, ‘R’ be the radius of earth, and ‘h’ be the height from which the object is to be projected.
Suppose that the object is thrown with initial velocity vi and reaches the infinity with final velocity of Vf. Then the total energy of object at infinity can be given as,
But by the law of conservation of energy we can say,
When object reaches to infinite height final velocity becomes zero, then the above equation can be given as,
If the object is projected from the surface of earth, then h = 0
This is the minimum velocity required to project the object so that it can escape from the surface of earth, known as escape velocity.
We know that gR2=GM
Points to remember:
- Escape velocity is independent of mass of object
- It depends upon the mass and radius of planet.
Numerical based on escape speed
Ex 1: Find the escape velocity for an object from the surface of planet whose mass is 5 times that of earth and radius is 1/3rd that of earth.
Solution:
Ex 2: The radii of planet are in ratio 2:3 and their masses are in ratio 1:3, then find the ratio of escape speeds for the planets.
Solution: