$A$ man at the top of a tower throws an object horizontally,whereas he simply drops another. Will these two objects reach the earth at the same time?

$A$ girl stands on a box having $60 \,cm$ length,$40 \,cm$ breadth,and $20 \,cm$ width in three ways. In which of the following cases will the pressure exerted by the box be maximum?

The mass of the planet Jupiter is $1.9 \times 10^{27} \, kg$ and that of the Sun is $1.99 \times 10^{30} \, kg$. The mean distance of the Sun from Jupiter is $7.8 \times 10^{11} \, m$. Calculate the gravitational force which the Sun exerts on Jupiter.

$(a)$ $A$ cube of side $5\, cm$ is immersed in water and then in a saturated salt solution. In which case will it experience a greater buoyant force? If each side of the cube is reduced to $4\, cm$ and then immersed in water,what will be the effect on the buoyant force experienced by the cube as compared to the first case for water? Give reasons for each case.
$(b)$ $A$ ball weighing $4\, kg$ with a density of $4000\, kg\, m^{-3}$ is completely immersed in water of density $10^3\, kg\, m^{-3}$. Find the force of buoyancy on it. (Given $g = 10\, m\, s^{-2}$)

Write the expression for the acceleration due to gravity at the surface of the earth.

The acceleration due to gravity at the moon's surface is $1.67 \, m s^{-2}$. If the radius of the moon is $1.74 \times 10^{6} \, m$,calculate the mass of the moon. (Use $G = 6.67 \times 10^{-11} \, N m^{2} kg^{-2}$)