A stone is projected vertically up from the bottom of a water tank. Assuming no water resistance it will go up and come down in same time but if water drag is present then the time it takes to go up, $t_{up}$ and the time it takes to come down, $t_{down}$ are related as

An empty glass jar is submerged in tank of water with open mouth of the jar downwards, so that air inside the jar is trapped and cannot get out. As the jar is pushed down slowly, the magnitude of net buoyant force on the system of volume of gas trapped in the jar and the jar

A body floats in a liquid contained in a beaker. If the whole system as shown in figure falls freely under gravity, then the upthrust on the body due to liquid is

Karman line is a theoretical construct that separates the earth's atmosphere from outer space. It is defined to be the height at which the lift on an aircraft flying at the speed of a polar satellite $(8 \,km / s )$ is equal to its weight. Taking a fighter aircraft of wing area $30 \,m ^2$, and mass $7500 \,kg$, the height of the Karman line above the ground will be in the range ………….. $km$ (assume the density of air at height $h$ above ground to be $\rho( h )=1.2 e ^{\frac{ h }{10}} \,kg / m ^3$ where $h$ is in $km$ and the lift force to be $\frac{1}{2} \rho v^2 A$, where $v$ is the speed of the aircraft and $A$ its wing area).

A cone of radius $R$ and height $H$, is hanging inside a liquid of density $\rho$ by means of a string as shown in the figure. The force, due to the liquid acting on the slant surface of the cone is