derive the expression for escape velocity.
Answers
Answer:
escape velocity is the minimum velocity with which a body must be projected vertically
upward so that it may just escape the surface of
the earth. Expression for escape velocity let a body of mass m be escaped from the gravitational field of the earth.
Answer:
Explanation:
Now, let the minimum velocity required from the body to escape the planet be-> ve
Thus, the 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 VelocityNow, 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 = 9.8m/
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.
This was the derivation of the escape velocity of earth or any other planet. This escape velocity derivation is very crucial as questions related to this topic are common in the physics exams.