A ball is thrown from ground at an angle $$\theta $$ with horizontal and with an initial speed u₀. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V₁. After hitting the ground, the ball rebounds at the same angle $$\theta $$ but with a reduced speed of $${{{u_0}} \over \alpha }$$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V₁, the value of $$\alpha $$ is ..................

A ball is projected from the ground at an angle of $${45^o}$$ with the horizontal surface. It reaches a maximum height of $$120$$ $$m$$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $${30^o}$$ with the horizontal surface. The maximium height it reaches after the bounce, in metres, is ______________.

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