Final answer:
The time taken for the boy to walk up on the moving escalator can be found using the harmonic mean of the times t1 and t2.
Step-by-step explanation:
The time taken by the boy to walk up on the moving escalator can be found by combining the rates of the boy walking and the escalator moving. When he walks up the stationary escalator, it takes him time t1. When he stands still on the moving escalator, it takes time t2 to reach the top. To calculate the time taken to walk up the moving escalator, we let the speed of the boy be v and the speed of the escalator be e. Therefore, his total speed on the moving escalator is v+e, and the time taken t can be determined by considering the harmonic mean of t1 and t2 because the total distance covered by the boy is the same in each scenario.
When walking up a stationary escalator, the boy's speed is equal to the speed of the escalator. Let's say the length of the escalator is 'd' and the boy's walking speed is 'v'. Therefore, the time taken by the boy to walk up the stationary escalator (t1) is given by t1 = d/v.
When the boy remains stationary and the escalator is moving, the time taken by the escalator to take him up (t2) is also d/v, as the escalator covers the entire length d with speed v.
Therefore, the time taken by the boy to walk up on the moving escalator is t1 + t2 = d/v + d/v = 2d/v.