A ball is thrown from ground at an angle $\mathrm{\theta }$ with horizontal and with an initial speed ${\mathrm{u}}_0.$ 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 ${\mathrm{V}}_1.$ After hitting the ground, ball rebounds at the same angle $\mathrm{\theta }$ but with a reduced speed of $\frac{{\mathrm{u}}_0}{\mathrm{\alpha }}.$ 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 $0.8{\mathrm{V}}_1,$ the value of $\mathrm{\alpha }$ is______

For first projectile
Average velocity, \(\langle V\rangle=\frac{R}{T}=U_x=V_1\) \(\left[\begin{array}{c}(\text { where } R=\text { Range and } T=\text { Time of flight }) \\ R=\frac{2 U_x \cdot U_y}{g} \quad T=\frac{2 \cdot U_y}{g}\end{array}\right]\)
For journey
\(\langle V\rangle_{1 \rightarrow n}=\frac{R_1+R_2+\ldots+R_n}{T_1+T_2+\ldots+T_n}=\frac{\frac{2 u_{x_1} u_{y_1}}{g}+\frac{2 u_{x_2} u_{y_2}}{g}+\ldots+\frac{2 u_{x_n} u_{y_n}}{g}}{\frac{2 u_{y_1}}{g}+\frac{2 u_{y_2}}{g}+\ldots \frac{2 u_{y_n}}{g}}\)
\(\Rightarrow U_x\left[\frac{1+\frac{1}{\alpha^2}+\frac{1}{\alpha^4}+\ldots \frac{1}{\alpha^{2 n}}}{1+\frac{1}{\alpha}+\frac{1}{\alpha^2}+\ldots+\frac{1}{\alpha^n}}\right]=0.8~ v_1,\)\(\left[\begin{array}{c}1+\frac{1}{\alpha^2}+\frac{1}{\alpha^4}+\ldots \frac{1}{\alpha^{2 n}} \Rightarrow \text { Geometric progression } \\ \text { Sum of G.P. }=\frac{1}{1-\alpha^2}\end{array}\right]\)
\(\Rightarrow \frac{V_1\left[\frac{1}{1-\frac{1}{\alpha^2}}\right]}{\left[\frac{1}{1-\frac{1}{\alpha}}\right]}=0.8 v_1 \Rightarrow \frac{\alpha}{1+\alpha}=0.8 \Rightarrow \alpha=4\)