Airplanes A and B are flying with constant velocity in the same vertical plane at angles $$30^\circ $$ and $$60^\circ $$ with respect to the horizontal respectively as shown in the figure. The speed of A is $$100\sqrt 3 $$ m/s. At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at t = t₀, A just escapes being hit by B, t₀ in seconds is

A rocket is moving in a gravity free space with a constant acceleration of 2 ms^–2 along + x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in + x direction with a speed of 0.3 ms^–1 relative to the rocket. At the same time, another ball is thrown in - x direction with a speed of 0.2 ms^–1 from its right end relative to the rocket. The time in seconds when the two balls hit each other is

Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n × 10) Joules. The value of n is (take acceleration due to gravity = 10 ms^–2)

Let $${E_1}\left( r \right),{E_2}\left( r \right)$$ and $${E_3}\left( r \right)$$ be the respective electric field at a distance $$r$$ from a point charge $$Q,$$ an infinitely long wire with constant linear charge density $$\lambda ,$$ and an infinite plane with uniform surface charge density $$\sigma .$$ If $$E{}_1\left( {{r_0}} \right) = {E_2}\left( {{r_0}} \right) = {E_3}\left( {{r_0}} \right)$$ at a given distance $${r_0}.$$ then