Answer

Verified

428.1k+ views

**Hint :**Minimum speed with which a body has to be projected vertically upwards from the surface of the earth (or any other planet) so that it just crosses the gravitational field of earth and never returns on its own is known as ESCAPE SPEED.

As the speed of the projection is increased, the body rises up to a certain height and then falls back to earth. As the speed of projection is increased the body attains a greater height before falling. Finally, a stage reaches, when the speed is so large that it just crosses the gravitational field and will never return back to the earth on its own. The body is said to have escaped.

**Complete step by step answer**

Let earth be a perfect sphere of mass M, radius R, with centre at O. let a body of mass m to be projected from a point A on the surface of earth. Join OA and produce if further. Take two points P and Q at a distance x and (x + dx) from the center O of the earth.

Gravitational Force of attraction at P is:

$ F=\dfrac{GMm}{{{x}^{2}}} $

Work done in taking the body against gravitational attraction P and Q is:

$ \begin{align}

& dW=Fdx \\

& =\dfrac{GMm}{{{x}^{2}}}dx \\

\end{align} $

Total work done in limits (x = R to x = $ \infty $ ):

$ W=\int\limits_{R}^{\infty }{\dfrac{GMm}{{{x}^{2}}}}dx $

$ \Rightarrow GMm{{\left[ \dfrac{{{x}^{-2+1}}}{-2+1} \right]}^{\infty }}_{R} $

$ \Rightarrow -GMm{{\left[ \dfrac{1}{x} \right]}^{\infty }}_{R} $

$ \Rightarrow -GMm\left[ \dfrac{1}{\infty }-\dfrac{1}{R} \right] $

$ W=\dfrac{GMm}{R} $

This work done is at the cost of K.E. given to the body at the surface of the earth. If $ {{v}_{e}} $ is the escape speed of the body projected from the surface of the earth, then

K.E. of the body = $ \dfrac{\text{1}}{\text{2}}\text{m}{{\text{v}}_{\text{e}}}^{\text{2}} $

$ \dfrac{1}{2}m{{v}_{e}}^{2}=\dfrac{GMm}{R} $

Or

$ {{v}_{e}}^{2}=\dfrac{2GM}{R} $

$ {{v}_{e}}=\sqrt{\dfrac{2GM}{R}} $

**Therefore option (A) is correct.**

**Additional Information**

When the body goes just out of the gravitational field of the earth, then the total energy of the body at infinity is zero. Now $ {{v}_{e}} $ is called escape speed. Using law of conservation of mechanical energy for the body;

(P.E. + K.E) of the body at the surface of earth = total energy at infinity

$ \Rightarrow -\dfrac{GMm}{R}+\dfrac{1}{2}m{{v}_{e}}^{2}=0 $

$ \Rightarrow {{v}_{e}}^{2}=\dfrac{2GM}{R} $

$ \Rightarrow {{v}_{e}}=\sqrt{\dfrac{2GM}{R}} $

**Note**

The value of escaped speed does not depend upon the mass (m) of the body and its angle of projection from the surface of earth or planet. it depends on the mass and radius of the planet from the surface of which the body is to be projected. If a body is projected from a planet with a speed v which is smaller than the escape speed, then the body will reach a certain height may either move in an orbit around the planet or may fall back to the planet. Make sure while projecting a body, the body should have higher speed as compared to escape speed.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE