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Prove that the escape velocity from the surface of the earth is 11.2 km/s

Answer» Assume a perfect sphere-shaped planet of radius\xa0R\xa0and mass\xa0M. Now, if a body of mass\xa0m\xa0is projected from a point\xa0A\xa0on the surface of the planet. An image is given below for better representation:sIn the diagram, a line from the center of the planet i.e.\xa0O\xa0is drawn till\xa0A\xa0(OA) and extended further away from the surface. In that extended line, two more points are taken as\xa0P\xa0and\xa0Q\xa0at a distance of\xa0x\xa0and\xa0dx\xa0respectively from the center O.Now, let the minimum velocity required from the body to escape the planet bveThus, Kinetic Energy will beAt point\xa0P, the body will be at a distance\xa0x\xa0from the planet’s center and the force of gravity between the object and the planet will be:To take the body from\xa0P\xa0to\xa0Q\xa0i.e. against the gravitational attraction, the work done will be->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\xa0x=R\xa0to\xa0x=∞.Thus,By integrating it further, the following is obtained:Thus, the work done will be: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. = WPutting the value for K.E. and Work, the following equation is obtained:From this equation, the escape velocity can be easily formulated which is:Putting the value of\xa0g = GM,\xa0the value of escape velocity becomes: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\xa0mR = 63,781,00 mSo, the escape velocity will be:ve=2×9.8×63,781,00−−−−−−−−−−−−−−−−√Escape Velocity of Earth= 11.2 km/s.


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