(A) The energy required to launch an object of mass $m$ from the earth's surface to infinity (free space) is equal to the gravitational potential ener

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(A) The energy required to launch an object of mass $m$ from the earth's surface to infinity (free space) is equal to the gravitational potential energy at the surface,which is given by the formula: $E = \frac{GMm}{R}$.Since the acceleration due to gravity at the surface is $g = \frac{GM}{R^2}$,we can write $GM = gR^2$.Substituting this into the energy equation: $E = \frac{(gR^2)m}{R} = mgR$.Given values: $m = 1000 \ kg$,$g = 10 \ m/s^2$,and $R = 6400 \ km = 6.4 \times 10^6 \ m$.Calculating the energy: $E = 1000 \times 10 \times 6.4 \times 10^6 = 6.4 \times 10^{10} \ J$.

Class 11 Physics Gravitation Escape Velocity and Escape Energy

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The mass of a spaceship is $1000 \ kg$. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10 \ m/s^2$ and $6400 \ km$ respectively. The required energy for this work will be

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A
$6.4 \times 10^{10} \ J$
B
$6.4 \times 10^{11} \ J$
C
$6.4 \times 10^8 \ J$
D
$6.4 \times 10^9 \ J$

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Escape Velocity and Escape Energy

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The ratio of accelerations due to gravity $g_{1}:g_{2}$ on the surfaces of two planets is $5:2$ and the ratio of their respective average densities $\rho_{1}:\rho_{2}$ is $2:1$. What is the ratio of respective escape velocities $v_{1}:v_{2}$ from the surface of the planets?

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$A$ rocket is projected in the vertically upwards direction with a velocity $kv_e$,where $v_e$ is the escape velocity and $k < 1$. The distance from the centre of the Earth up to which the rocket will reach is:

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Match Column-$I$ with Column-$II$.

Column-$I$ Column-$II$

$(1)$ Value of escape velocity on the surface of the Earth $(a)$ $2.38 \, km \, s^{-1}$

$(2)$ Value of escape velocity on the surface of the Moon $(b)$ $7.92 \, km \, s^{-1}$

$(c)$ $11.2 \, km \, s^{-1}$

Column-$I$	Column-$II$
$(1)$ Value of escape velocity on the surface of the Earth	$(a)$ $2.38 \, km \, s^{-1}$
$(2)$ Value of escape velocity on the surface of the Moon	$(b)$ $7.92 \, km \, s^{-1}$
	$(c)$ $11.2 \, km \, s^{-1}$

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The escape velocity from the Earth's surface is $v$. The escape velocity from the surface of another planet having a radius four times that of Earth and the same mass density is:

MediumNEET 2021

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If a body of mass $1\text{ kg}$ falls on the earth from infinity, it attains velocity $(v)$ and kinetic energy $(k)$ on reaching the surface of earth. The values of $v$ and $k$ respectively are . . . . . . . (Take radius of earth to be $6400\text{ km}$ and $g = 9.8\text{ m/s}^2$)

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The mass of a spaceship is $1000 \ kg$. It is to be launched from the earth's surface out into free space. The value of $g$ and $R$ (radius of earth) are $10 \ m/s^2$ and $6400 \ km$ respectively. The required energy for this work will be

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The ratio of accelerations due to gravity $g_{1}:g_{2}$ on the surfaces of two planets is $5:2$ and the ratio of their respective average densities $\rho_{1}:\rho_{2}$ is $2:1$. What is the ratio of respective escape velocities $v_{1}:v_{2}$ from the surface of the planets?

$A$ rocket is projected in the vertically upwards direction with a velocity $kv_e$,where $v_e$ is the escape velocity and $k < 1$. The distance from the centre of the Earth up to which the rocket will reach is:

Match Column-$I$ with Column-$II$. Column-$I$ Column-$II$ $(1)$ Value of escape velocity on the surface of the Earth $(a)$ $2.38 \, km \, s^{-1}$ $(2)$ Value of escape velocity on the surface of the Moon $(b)$ $7.92 \, km \, s^{-1}$ $(c)$ $11.2 \, km \, s^{-1}$

The escape velocity from the Earth's surface is $v$. The escape velocity from the surface of another planet having a radius four times that of Earth and the same mass density is:

If a body of mass $1\text{ kg}$ falls on the earth from infinity, it attains velocity $(v)$ and kinetic energy $(k)$ on reaching the surface of earth. The values of $v$ and $k$ respectively are . . . . . . . (Take radius of earth to be $6400\text{ km}$ and $g = 9.8\text{ m/s}^2$)

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Match Column-$I$ with Column-$II$.

Column-$I$ Column-$II$

$(1)$ Value of escape velocity on the surface of the Earth $(a)$ $2.38 \, km \, s^{-1}$

$(2)$ Value of escape velocity on the surface of the Moon $(b)$ $7.92 \, km \, s^{-1}$

$(c)$ $11.2 \, km \, s^{-1}$