Derive the formula for escape velocity in terms of acceleration due to gravity and radius of the
planet.
Answers
Answer:
Explanation:
Derivation of Escape Velocity:
Before checking the escape velocity derivation, it is important to know what escape velocity is and what are its related concepts. Check out Escape Velocity of Earth for a detailed understanding of the minimum velocity required to overcome the earth’s gravitational pull.
Escape Velocity Formula:
ve=√2gR
Derivation:
Assume a perfect sphere-shaped planet of radius R and mass M. Now, if a body of mass m is projected from a point A on the surface of the planet. An image is given below for better representation:
Derivation of Escape Velocity
In the diagram, a line from the center of the planet i.e. O is drawn till A (OA) and extended further away from the surface. In that extended line, two more points are taken as P and Q at a distance of x and dx respectively from the center O.
Now, let the minimum velocity required from the body to escape the planet b
ve
Thus, Kinetic Energy will be
Derivation of Escape Velocity
At point P, the body will be at a distance x from the planet’s center and the force of gravity between the object and the planet will be:
Derivation of Escape Velocity
To take the body from P to Q i.e. against the gravitational attraction, the work done will be->
Derivation of Escape Velocity
Now, the work done against the gravitational attraction to take the body from the planet’s surface to infinity can be easily calculated by integrating the equation for work done within the limits x=R to x=∞.
Thus,
Derivation of Escape Velocity
By integrating it further, the following is obtained:
Derivation of Escape Velocity
Thus, the work done will be:
Derivation of Escape Velocity
Now, to escape from the surface of the planet, the kinetic energy of the body has to be equal to the work done against the gravity going from the surface to infinity. So,
K.E. = W
Putting the value for K.E. and Work, the following equation is obtained:
Derivation of Escape Velocity
From this equation, the escape velocity can be easily formulated which is:
Derivation of Escape Velocity
Putting the value of g = GM, the value of escape velocity becomes:
Derivation of Escape Velocity
From this equation, it can be said that the escape velocity depends on the radius of the planet and the mass of the planet only and not on the mass of the body.
Escape Velocity of Earth:
From the above equation, the escape velocity for any planet can be easily calculated if the mass and radius of that planet are given. For earth, the values of g and R are:
g = 9.8 m
R = 63,781,00 m
So, the escape velocity will be:
ve=√2×9.8×63,781,00
Escape Velocity of Earth= 11.2 km/s.