Explanation

(b) According to given problem $\frac{1}{2}mv^{2} = as^{2}$
$\Rightarrow v = s\sqrt{\frac{2a}{m}}$
$So\ a_{R} = \frac{v^{2}}{R} = \frac{2as^{2}}{mR}$ …(i)
Further more as
$a_{t} = \frac{dv}{dt} = \frac{dv}{ds}.\frac{ds}{dt} = v\frac{dv}{ds}$ …(ii)
[By chain rule]
Which is light of equation (i) i.e. $v = s\sqrt{\frac{2a}{m}}$ yields
$a_{t} = \left\lbrack s\sqrt{\frac{2a}{m}} \right\rbrack\left\lbrack \sqrt{\frac{2a}{m}} \right\rbrack = \frac{2as}{m}$ …(iii)
So that $a = \sqrt{a_{R}^{2} + a_{t}^{2}} = \sqrt{\left\lbrack \frac{2as^{2}}{mR} \right\rbrack^{2}\left\lbrack \frac{2as}{m} \right\rbrack^{2}}$
Hence $a = \frac{2as}{m}\sqrt{1 + \lbrack s/R\rbrack^{2}}$
$\therefore F = ma = 2as\sqrt{1 + \lbrack s/R\rbrack^{2}}$