Question

The total mechanical energy of an object of mass m projected surface of Earth with escape speed is

Answer 1

The escape speed is $\sqrt{v^2-gR}$

Explanation:

Given that,

Mass of object = m

Suppose the escape speed of particle at height, h = R

During motion of the particle, total mechanical energy remains constant.

The total mechanical energy at the surface of earth is

$E_{s}=K.E_{s}+P.E_{s}$

Put the value into the formula

$E_{s}=\dfrac{1}{2}mv^2-\dfrac{GmM}{R}$......(I)

Where, m = mass of object

M = Mass of earth

R = radius of earth

The total mechanical energy at the height of earth surface is

$E_{h}=K.E_{h}+P.E_{h}$

Put the value into the formula

$E_{h}=\dfrac{1}{2}mv'^2-\dfrac{GmM}{2R}$......(II)

We need to clculate the escape speed

Using conservation of energy

From equation (I) and (II)

$\dfrac{1}{2}mv^2-\dfrac{GmM}{R}=\dfrac{1}{2}mv'^2-\dfrac{GmM}{2R}$...(III)

We know that,

The gravitational force is

$g=\dfrac{GM}{R^2}$

Put the value in equation (III)

$\dfrac{1}{2}mv^2-gmR=\dfrac{1}{2}mv'^2-\dfrac{gmR}{2}$

$v'=\sqrt{v^2-gR}$

Hence, The escape speed is $\sqrt{v^2-gR}$

Learn more :

Topic : escape velocity

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Answer 2

Hey Dhruv! Thanks for putting your doubt.

Question esquires about the value of total mechanical energy of an object of mass projected from surface of the earth with the escape speed.

We know from the concept of escape speed that is any body is projected with this speed from a surface, it is bound to travel till infinity pulling itself out of the gravitational effects of the planetary body.

Applying conservation of mechanical energy we can say that the total mechanical energy at the infinity is going to result out to zero considering the kinetic energy of the projected body and gravitational potential energy.

Hence, the total mechanical energy at infinity will be zero.