A particle of mass m is projected with a speed u from the ground at an angle $$\theta = {\pi \over 3}$$ w.r.t. horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $$u\widehat i$$ . The horizontal distance covered by the combined mass before reaching the ground is:

A particle starts from the origin at t = 0 with an
initial velocity of 3.0 $$\widehat i$$ m/s and moves in the
x-y plane with a constant acceleration $$\left( {6\widehat i + 4\widehat j} \right)$$ m/s² . The x-coordinate of the particle at the instant when its y-coordinate is 32 m is D meters. The value of D is :-

There is a small source of light at some depth below the surface of water (refractive index = $${4 \over 3}$$) in a tank of large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly) :
[Use the fact that surface area of a spherical cap of height h and radius of curvature r is 2$$\pi $$rh]:

A small circular loop of conducting wire has radius a and carries current I. It is placed in a uniform magnetic field B perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period T. If the mass of the loop is m then :