An insulated container contains a monoatomic gas of molar mass $M$. The container is moving with velocity $V$. If it is stopped suddenly, the change in temperature of the gas is ($R$ = gas constant).

$A$ rocket is propelled by a gas which is initially at a temperature of $4000\, K$. The temperature of the gas falls to $1000\, K$ as it leaves the exhaust nozzle. The gas which will acquire the largest momentum while leaving the nozzle is:

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$. Assertion $A$: If the average kinetic energy of $H_2$ and $O_2$ molecules, kept in two different sized containers are same, then their temperatures will be same. Reason $R$: The r.m.s. speed of $H_2$ and $O_2$ molecules are same at same temperature. Choose the correct answer from the options given below.

$A$ very tall vertical cylinder is filled with a gas of molar mass $M$ under isothermal conditions at temperature $T$. The density and pressure of the gas at the base of the container are $\rho_0$ and $p_0$,respectively. Choose the correct statement$(s)$ if gravity is assumed to be constant throughout the container.

Consider the following statements for air molecules in an air-tight container:
$(I)$ The average speed of molecules is larger than root mean square speed.
$(II)$ Mean free path of molecules is larger than the mean distance between molecules.
$(III)$ Mean free path of molecules increases with temperature.
$(IV)$ The rms speed of nitrogen is smaller than oxygen molecule.
Which of the above statements are correct?

Two closed containers of equal volume are filled with air at pressure $P_0$ and temperature $T_0$. Both are connected by a narrow tube. If one of the containers is maintained at temperature $T_0$ and the other at temperature $T$,then the new pressure in the containers will be: