A rocket is launched straight up from the surface of the earth. When its altitude is one fourth of the radius of the earth, its fuel runs out and therefore it coasts. The minimum velocity which the rocket must have when it starts to coast if it is to escape from the gravitational pull of the earth is [escape velocity on surface of earth is $11.2 k m / s$ ]

Question

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Answer 1

Let $R$ be the radius of Earth and $m$ be the mass of the rocket.
At the altitude $\frac{R}{4}$ from surface of Earth:
Let $v$ be the total speed of the rocket at that altitude of $\frac{R}{4}$ . This speed is inclusive of the angular velocity of Earth's rotation and due to the propelling engine.

$K . E = \frac{1}{2} m v^{2}$

$P . E = - \frac{G M m}{( 1 . 2 5 R )}$

While the object is inside the Earth's gravitational field, the total energy is negative. For the rocket to escape the Earth's gravitational field, the total energy must be 0 or more. Then at infinite distance the energy of the rocket will be 0 or more. So it will not come back towards Earth.

$\frac{1}{2} m v^{2} = \frac{G M m}{( 1 . 2 5 R )}$

$v = \frac{2 G M}{R} \times 0.8$

$= 11.2 k m / s e c \times 0.9 \approx 10 k m / s$

Answer 2

Let

R

be the radius of Earth and

m

be the mass of the rocket.

At the altitude

\frac{R}{4}

from surface of Earth:

Let

v

be the total speed of the rocket at that altitude of

\frac{R}{4}

. This speed is inclusive of the angular velocity of Earth's rotation and due to the propelling engine.

K . E = \frac{1}{2} m v^{2}

P . E = - \frac{G M m}{( 1 . 2 5 R )}

While the object is inside the Earth's gravitational field, the total energy is negative. For the rocket to escape the Earth's gravitational field, the total energy must be 0 or more. Then at infinite distance the energy of the rocket will be 0 or more. So it will not come back towards Earth.

\frac{1}{2} m v^{2} = \frac{G M m}{( 1 . 2 5 R )}

v = \frac{2 G M}{R} \times 0.8

= 11.2 k m / s e c \times 0.9 \approx 10 k m / s

$1 k m / s$

$5 k m / s$

$10 k m / s$

$15 k m / s$

A satellite is moving with an orbital velocity u at a distance R from the surface of the earth. With what velocity should it move, so that it escapes the gravitational pull of the earth. R is the radius of the earth

For a planet having mass equal to mass of the earth but radius is one fourth of radius of the earth. Then escape velocity for this planet will be

A spaceship goes into a circular orbit close to the earth's surface. What additional velocity must be imparted to the ship so that it is able to escape the gravitational pull of the earth? (R $= 6400$ km, g $= 9.8$ m $s^{- 2}$ )

The escape velocity from an altitude equal to radius of the Earth above Earth's surface is:
$($ Escape velocity from surface of the Earth is $11.2$ kms $^{- 1}$ $)$

1 km/s

5 km/s

10 km/s

15 km/s