What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ into a circular orbit at an altitude of $2R$?

Which of the following has the lowest melting point?

Two particles of equal mass have velocities $\overrightarrow{v}_1=4 \hat{i} \text{ ms}^{-1}$ and $\overrightarrow{v}_2=4 \hat{j} \text{ ms}^{-1}$. The first particle has an acceleration $\overrightarrow{a}_1=(5 \hat{i}+5 \hat{j}) \text{ ms}^{-2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of

In a playground,there is a merry-go-round of mass $120 \ kg$ and radius $4 \ m$. The radius of gyration is $3 \ m$. $A$ child of mass $30 \ kg$ runs at a speed of $5 \ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.

The rearrangement of phenyl esters to $o-$ and $p-$ hydroxy ketones in the presence of $AlCl_3$ is known as:

If the image of $\left(\frac{-7}{5}, \frac{-6}{5}\right)$ in a line is $(1, 2)$,then the equation of the line is