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**JEE Mains**

Planet $A$ has mass $M$ and radius $R$. Planet $B$ has half the mass and half the radius of planet $A$. If the escape velocities from the planets $A$ and $B$ are $υ_{A}$ and $υ_{B}$ respectively, then $υ_{B}υ_{A} =4n $.

The value of $n$ is

An asteroid is moving directly towards the centre of the earth. When at a distance of $10R$ ($R$ is the radius of the earth) from the earth centre, it has a speed of $12km/s$. Neglecting the effect of the earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is $11.2km/s$)? Give your answer to the nearest integer in kilometer/s _____

Consider two solid spheres of radii $R_{1}=1m,R_{2}=2m$ and masses $M_{1}$ and $M_{2}$, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of $M_{2}M_{1} $ is:

A box weights $196N$ on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $g=10ms_{−2}$ at the north pole and the radius of the earth $=6400km$):

A satellite of mass $m$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After it reaches height R(R= radius of the earth), it ejects a rocket of mass $10m $ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ($G$ is the gravitational constant; $M$ is the mass of the earth) :

The ratio of the weights of a body on the Earth's surface to that on the surface of a planet is $9:4$. The mass of the planet is $91 th$ of that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet ?

(Take the planets to have the same mass density)

(Take the planets to have the same mass density)

A test particle is moving in a circular orbit in the gravitational field produced by a mass density $ρ(r)=r_{2}K $. Identify the correct relation between the radius R of the particle's orbit and its period T.

A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon ? Assume that the density of the earth and the moon are equal and that the earth's volume is 64 times the volume of the moon :

Four identical particles of mass $M$ are located at the corners of a square of side $_{′}a_{′}$. What should be their speed if each of them revolves under the influence of other's gravitational field in a circular orbit circumscribing the square?

The variation of acceleration due to gravity $g$ with distance $d$ from centre of the earth is best represented by (R=Earth's radius) :

A satellite is revolving in a circular orbit at a height 'h' from the earth's surface(radius of earth R$;$ h$<<$R). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field, is closed to(Neglect the effect of atmosphere.)