Question

One has to throw a particle from one side of a fixed sphere, in diametrical plane to another side such that it just grazes the sphere. Minimum possible speed for this is $\sqrt{2gR\left(\sqrt{\alpha }+\sqrt{\beta }\right)}.$Then $\alpha +\beta$ is. Write upto two digits after the decimal point.

Answer 1

$\frac{2v^{2}sin\theta cos\theta }{g}=2Rsin\theta \Rightarrow v^2=\frac{Rg}{cos\theta }$
$u^2=v^2+2gR(1+cos\theta )$
$u^2=\frac{Rg}{cos\theta }+2gR+2gRcos\theta$
$u^2=Rg\left(\frac{1+2co{s}^2\theta }{cos\theta }\right)+2gR$
for u to be minimum $\frac{1+2co{s}^2\theta }{cos\theta }=min$
$\Rightarrow cos\theta =\frac{1}{\sqrt{2}}\Rightarrow \theta =\frac{\pi }{4}$
$u_{min}=\sqrt{\sqrt{2}Rg+2gR+\sqrt{2}Rg}=\sqrt{2gR(\sqrt{2}+1)}$