A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet’s gravity is $${\left( {{1 \over 4}} \right)^{th}}$$ of its value at the surface of the planet. If the escape velocity from the planet is $${v_{esc}} = v\sqrt N $$, then the value of N is (ignore energy loss due to atmosphere)

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature T. Assuming the gases are ideal, the correct statement(s) is(are)

Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits 10⁴ times the power emitted from B. The ratio $$\left( {{{{\lambda _A}} \over {{\lambda _B}}}} \right)$$ of their wavelengths $${{\lambda _A}}$$ and $${{\lambda _B}}$$ at which the peaks occur in their respective radiation curves is

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $$\lambda $$ are kept parallel to each other. In their resulting electric field, point charges $$q$$ and $$-q$$ are kept in equilibrium between them. The point charges are confined to move in the $$x$$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is (are)