A body is projected vertically upward from the surface of the earth with a velocity equal to half its escape velocity. If R is radius of the earth, then the maximum height attained by the body is (R / n). Find n.

Question

A body is projected vertically upward from the surface of the earth with a velocity equal to half its escape velocity. If R is radius of the earth, then the maximum height attained by the body is (R / n). Find n. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let the body be projected from the surface of the earth with velocity

v

and reaches maximum height

h

.
According to law of conservation of mechanical energy, we get

\frac{1}{2} m v^{2} - \frac{GM m}{R} = 0 - \frac{GM m}{( R + h )}

or

\frac{1}{2} v^{2} - \frac{GM}{R} = - \frac{GM}{( R + h )}

As per question,

V = \frac{V _{e}}{2} = \frac{1}{2} \frac{2 GM}{R}

[∵ v_{e} = \frac{2 GM}{R}]

\Rightarrow \frac{1}{2} (\frac{1}{2} \frac{GM}{R}) - \frac{GM}{R} = - \frac{GM}{( R + h )}

\frac{GM}{4 R} - \frac{GM}{R} = - \frac{GM}{R + h}

- \frac{3}{4 R} = - \frac{1}{( R + h )}

3 (R + h) = 4 R

or

3 R + 3 h = 4 R

3 h = R

or

h = \frac{R}{3}

Q. A body is projected vertically upward from the surface of the earth with a velocity equal to half its escape velocity. If R is radius of the earth, then the maximum height attained by the body is (R/n). Find n.

Q. A body is projected vertically upward from the surface of the earth with a velocity equal to half its escape velocity. If $R$ is radius of the earth, then the maximum height attained by the body is $(R / n)$ . Find $n$ .