Question

A body is projected from earth with a velocity equal to 3 times escape velocity from Earth ,find its velocity at a distance far away from Earth?if the body is projected with a velocity equal to half of escape velocity from earth ,how much maximum height the body will reach above the earth?

Answer 1

The velocity at a distance far away from Earth is √8 times the escape velocity

The maximum height that the body will reach above the Earth will be the one third of the radius of the Earth

Explanation:

If the escape velocity of the body is $v_e$ then

The initial velocity at the surface of the earth = $3v_e$

The escape velocity if the velocity with which the object just escapes the gravitational field of the earth

Thus,

$\frac{1}{2}mv_e^2=\frac{GMm}{R}$

Where M is the mass of the earth, R is the radius of the earth and m is the mass of the body

or, $v_e=\sqrt{\frac{2GM}{R}}$

The gravitational potential energy at the surface of the earth

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

The total energy at the surface of the earth

$E=\frac{1}{2}m(3v_e)^2-\frac{GMm}{R}$

$\implies E=\frac{9}{2}mv_e^2-\frac{1}{2}mv_)e^2$

$\implies E=4mv_e^2$

At a distance far away from earth, the potential energy will be zero

If the velocity of the object there is v then

Total energy far away from earth

$E'=\frac{1}{2}mv^2$

Equating both energies

$4mv_e^2=\frac{1}{2}mv^2$

$\implies v=\sqrt{8}v_e$

Therefore, the velocity of the body far away from earth will be √8 times the escape velocity on earth.

If the body is projected with half of escape velocity then

Assuming it attains height R' from the centre of the earth

From energy conservation

$\frac{1}{2}m(\frac{v_e^2}{2})^2-\frac{GMm}{R}=0-\frac{GMm}{R'}$

$\implies \frac{1}{8}\times mv_e^2-\frac{GMm}{R}=-\frac{GMm}{R'}$

$\implies \frac{1}{8}\times \frac{2GMm}{R}-\frac{GMm}{R}=-\frac{GMm}{R'}$

$\implies -\frac{3GMm}{4R}=0-\frac{GMm}{R'}$

$\implies R'=\frac{4R}{3}$

The height attained above the surface of the earth

$h=R'-R$

$\implies h=\frac{4R}{3}-R$

$\implies h=\frac{R}{3}$

Therefore, the body will reach the height above the surface equal to one third of the radius of the earth.

Hope this answer is helpful.

Know More:

Q: A particle is thrown with escape velocity v from the surface of earth calculate its velocity at height 3R

Click Here: https://brainly.in/question/8011573