$A$ particle is dropped on Earth from a height $R$ (radius of Earth) and it bounces back to a height $R/2$. The coefficient of restitution for the collision is (ignore air resistance and rotation of Earth).

$A$ man of mass $m$ starts falling towards a planet of mass $M$ and radius $R.$ As he reaches near to the surface,he realizes that he will pass through a small hole in the planet. As he enters the hole,he sees that the planet is really made of two pieces: a spherical shell of negligible thickness of mass $\frac{2M}{3}$ and a point mass $\frac{M}{3}$ at the centre. The change in the force of gravity experienced by the man is:

The figure shows the orbit of a planet $P$ around the sun $S.$ $AB$ and $CD$ are the minor and major axes of the ellipse,respectively. If $U$ is the potential energy and $K$ is the kinetic energy of the planet,then $|U| > |K|$ at:

Two satellites of masses $m$ and $1.5 m$ are revolving around the earth in two circular orbits of heights $R_E$ and $2 R_E$ respectively,where $R_E$ is the radius of the earth. The ratio of the minimum and maximum gravitational forces exerted by the two satellites on the earth is

Given below are two statements: one is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$.
$Assertion$ $(A)$: The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun.
$Reason$ $(R)$: The moon takes less time to move around the earth than the time taken by the earth to move around the sun.
In the light of the above statements,choose the most appropriate answer from the options given below:

$A$ satellite of mass $5M$ orbits the Earth in a circular orbit. At one point in its orbit,the satellite explodes into two pieces,one of mass $M$ and the other of mass $4M$. After the explosion,the mass $M$ ends up travelling in the same circular orbit,but in the opposite direction. After the explosion,the mass $4M$ is in: