Question

the velocity with which a body should be projected from the surface of the earth such that it reaches a maximum height equal to n times the radius R of the earth

Answer 1

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Answer 2

Answer:

The velocity with which a body should be projected is $\sqrt{\frac{2GM}{R}\big[\frac{n}{n+1}\big]}$

Explanation:

Given a body is projected from the surface of the earth such that it reaches a maximum height equal to n times the radius R of the earth i.e., $h=nR$.

Let the velocity of the body projected is $v\frac{m}{s}$.

Kinetic energy of the body of mass m is, $KE=\frac{1}{2} mv^{2}$

Gravitational potential energy of the body at radius of earth R is given by

                         $PE=\frac{-GMm}{R}$

Given the maximum height , $h=nR$

Gravitational potential energy of the body at a height h is , $PE=\frac{-GMm}{nR+R}$

and at maximum height, $KE=0$

Following the principle of conservation of energy, $KE+PE=constant$,

                  $\frac{1}{2} mv^{2}+\frac{-GMm}{R}=\frac{-GMm}{nR+R}+0$

                            $\implies \frac{1}{2} v^{2}=\frac{GM}{R}-\frac{GM}{R(n+1)}$

                            $\implies \frac{1}{2} v^{2}=\frac{GM}{R}\big[\frac{n+1-1}{n+1}\big]$

                              $\implies v^{2}=\frac{2GM}{R}\big[\frac{n}{n+1}\big]$

                               $\implies v}=\sqrt{\frac{2GM}{R}\big[\frac{n}{n+1}\big]}$

Therefore, the maximum velocity with which a body should be projected from the surface of the earth is $\sqrt{\frac{2GM}{R}\big[\frac{n}{n+1}\big]}$