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Q.
A person walks up a stationary escalator in time $t_1$. If he remains stationary on the escalator, then it can take him up in time $t_2$. How much time would it take him to walk up the moving escalator?
Let $L$ be the length of escalator.
Speed of man w.r.t. escalator is $v_{mc} = \frac{L}{t_1}$
Speed of escalator is $v_c = \frac{L}{t_2}$
$ \therefore $ Speed of man with respect to ground would be
$ v_m = v_{mc} + v_c = L \left( \frac{1}{t_1} + \frac{1}{t_2} \right)$
$\therefore $ Time taken $t = \frac{L}{L\left(\frac{1}{t_{1}}+\frac{1}{t_{2}}\right)}=\frac{t_{1}t_{2}}{t_{1}+t_{2}}$