The mass of a planet is half that of the earth and the radius of the planet is one-fourth that of earth. If…

The mass of a planet is half that of the earth and the radius of the planet is one-fourth that of earth. If we plan to send an artificial satellite from the planet, the escape velocity will be, (escape velocity on earth $v_e=11 \mathrm{~km}-\mathrm{s}^{-1}$ )
  1. $11 \mathrm{~km}_{-\mathrm{s}^{-1}}$
  2. $5.5 \mathrm{~km}^{-\mathrm{s}^{-1}}$
  3. $15.55 \mathrm{~km}-\mathrm{s}^{-1}$
  4. $7.78 \mathrm{~km}-\mathrm{s}^{-1}$

Solution

Escape velocity $\begin{array}{rlrl} & \Rightarrow \quad v_{\mathrm{es}} & =\sqrt{\frac{2 G M}{R}} \\ \Rightarrow & & \frac{v_1}{v_2} & =\sqrt{\frac{M_1}{M_2} \times \frac{R_2}{R_1}} \\ \Rightarrow & & \frac{v_1}{v_2} & =\sqrt{\frac{M_1}{\frac{M_1}{2}} \times \frac{\frac{R_1}{4}}{R_1}}=\frac{1}{\sqrt{2}} \\ \Rightarrow & & v_2 & =\sqrt{2} \times 11=15.55 \mathrm{kms}^{-1}\end{array}$

Asked in: AP EAMCET 2007

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