Escape velocity

If a body of mass 1 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 km and $g=9.8$ m/s²)

Two spherical stars $A$ and $B$ have densities $\rho_A$ and $\rho_B$, respectively. $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B = 2M_A$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_A$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_A$. If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\dfrac{v_B}{v_A} = \sqrt{\dfrac{10n}{15^{1/3}}}$. The value of $n$ is _____.

The escape velocity of a body on a planet 'A' is 12 kms$^{-1}$. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is :

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be :

(Take radius of earth = 6400 km and g = 10 ms$^{-2}$)

A body of mass m is projected with velocity $\lambda v_e$ in vertically upward direction from the surface of the earth into space. It is given that $v_e$ is escape velocity and $\lambda<1$. If air resistance is considered to be negligible, then the maximum height from the centre of earth, to which the body can go, will be :

(R : radius of earth)

A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3 \times 10^5$ times heavier than the Earth and is at a distance $2.5 \times 10^4$ times larger than the radius of the Earth. The escape velocity from Earth’s gravitational field is $v_e = 11.2\ \mathrm{km\,s^{-1}}$. The minimum initial velocity $(v_S)$ required for the rocket to be able to leave the Sun-Earth system is closest to

(Ignore the rotation and revolution of the Earth and the presence of any other planet)

A bullet is fired vertically upwards with velocity $v$ from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet's gravity is 1/4th of its value at the surface of the planet. If the escape velocity from the planet is $v_{esc}=v\sqrt N$, then the value of $N$ is (ignore energy loss due to atmosphere)

Two bodies, each of mass $M$, are kept fixed with a separation $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is $G$. The correct statement(s) is (are)

Two spherical planets $P$ and $Q$ have the same uniform density $\rho$, masses $M_P$ and $M_Q$, and surface areas $A$ and $4A$, respectively. A spherical planet $R$ also has uniform density $\rho$ and its mass is $(M_P+M_Q)$. The escape velocities from the planets $P$, $Q$ and $R$, are $V_P$, $V_Q$ and $V_R$, respectively. Then

A satellite is moving with a constant speed '$V$' in a circular orbit about the earth. An object of mass '$m$' is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its ejection, the kinetic energy of the object is

Gravitational acceleration on the surface of a planet is $\dfrac{\sqrt6}{11}g$, where g is the gravitational acceleration on the surface of the earth. The average mass density of the planet is $\dfrac23$ times that of the earth. If the escape speed on the surface of the earth is taken to be 11 km s⁻¹, the escape speed on the surface of the planet in km s⁻¹ will be

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.