A ball is thrown from the location $\left(x_0, y_0\right)=(0,0)$ of a horizontal playground with an initial speed $v_0$ at an angle $\theta_0$ from the $+x$-direction. The ball is to be hit by a stone, which is thrown at the same time from the location $\left(x_1, y_1\right)=(L, 0)$. The stone is thrown at an angle $\left(180-\theta_1\right)$ from the $+x$-direction with a suitable initial speed. For a fixed $v_0$, when $\left(\theta_0, \theta_1\right)=\left(45^{\circ}, 45^{\circ}\right)$, the stone hits the ball after time $T_1$, and when $\left(\theta_0, \theta_1\right)=\left(60^{\circ}, 30^{\circ}\right)$, it hits the ball after time $T_2$. In such a case, $\left(T_1 / T_2\right)^2$ is ______.

A person of height $1.6 \mathrm{~m}$ is walking away from a lamp post of height $4 \mathrm{~m}$ along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60 \mathrm{~cm} \mathrm{~s}^{-1}$, the speed of the tip of the person's shadow on the ground with respect to the person is ____________ $\mathrm{cm}~ \mathrm{s}^{-1}$.

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 ..................