Two particles of equal mass m have respective
initial velocities $$u\widehat i$$ and $$u\left( {{{\widehat i + \widehat j} \over 2}} \right)$$.
They collide completely inelastically. The energy lost in the process is :

The aperture diameter of a telescope is 5m. The separation between the moon and the earth is 4 × 10⁵ km. With light of wavelength of 5500 $$\mathop A\limits^o $$, the minimum separation between objects on the surface of moon, so that they are just resolved, is close to :

A vessel of depth 2h is half filled with a liquid of refractive index $$2\sqrt 2 $$ and the upper half with another liquid of refractive index $$\sqrt 2 $$ . The liquids are immiscible. The apparent depth of the inner surface of the bottom of vessel will be :

A quantity f is given by $$f = \sqrt {{{h{c^5}} \over G}} $$ where c is speed of light, G universal gravitational constant and h is the Planck's constant. Dimension of f is that of :