An asteroid is moving directly towards the centre of the earth. When at a distance of $10 \mathrm{R}$ ($R$ is the radius of the earth) from the earths centre, it has a speed of $12 \;\mathrm{km} / \mathrm{s}$. Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is $11.2 \;\mathrm{km} / \mathrm{s}$ ) $?$

Give your answer to the nearest integer in $km/s$

A particle of mass $m$ moves around the origin in a potential $\frac{1}{2} m \omega^{2} r^{2}$, where $r$ is the distance from the origin. Applying the Bohr's model in this case, the radius of the particle in its $n$th orbit in terms of $a=\sqrt{h /(2 \pi m \omega)}$ is

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be $\frac{x}{5}\, \frac{ GM ^{2}}{ R }$ where $x$ is $\quad\,……….$ (Round off to the Nearest Integer)

($M$ is the mass of earth, $R$ is the radius of earth, $G$ is the gravitational constant)

The speed of a satellite in a circular orbit of radius $R_0$ around the earth is $v_0$. Another satellite is in an elliptic orbit around the earth. If the minimum and maximum speeds of the second satellite are $\alpha v_0$ and $\beta v_0$ respectively, then its time period is

Match list-$I$ with list-$II$:

(Where $\mathrm{a}=$ radius of planet orbit, $\mathrm{r}=$ radius of planet, $M=$ mass of Sun, $m=$ mass of planet)

Choose the correct answer from the options given below: