Question

A person standing on the escalator takes time \[{t_1}\] to reach the top of a tower when the escalator is moving. He takes time \[{t_2}\] to reach the top of the tower when the escalator is standing. How long will he take if he walks up a moving escalator?
(A) \[{t_2} - {t_1}\]
(B) \[{t_1} + {t_2}\]
(C) \[\dfrac{{{t_1}{t_2}}}{{{t_1} - {t_2}}}\]
(D) \[\dfrac{{{t_1}{t_2}}}{{{t_1} + {t_2}}}\]

Answer 1

Hint: Use the relation between the velocity, displacement and time to determine the speeds of escalator and person separately. Add these two speeds to obtain the combined velocity. Again, use the relation between the velocity, displacement and time to determine the time.

Formula used:
\[ \Rightarrow v = \dfrac{d}{t}\]
Here, d is the distance and t is the time.

Complete step by step answer:
Suppose \[d\] is the distance that the person wants to travel through the escalator to reach the top of the tower.
We know the relation between velocity, distance and time. The velocity of the body is the distance d divided by time t.
\[ \Rightarrow v = \dfrac{d}{t}\]
The speed of the escalator is,
\[{v_1} = \dfrac{d}{{{t_1}}}\]
Here, \[{t_1}\] is the time taken by the person to reach the top of the tower using a moving escalator.
The speed of the walking person over an escalator is,
\[ \Rightarrow {v_2} = \dfrac{d}{{{t_2}}}\]
Here, \[{t_2}\] is the time taken by the person to reach the top of the tower by walking.
The speed of the person while walking on the moving escalator is the sum of the above speeds.
Therefore,
\[ \Rightarrow v = \dfrac{d}{{{t_1}}} + \dfrac{d}{{{t_2}}}\]
\[ \Rightarrow v = d\left( {\dfrac{1}{{{t_1}}} + \dfrac{1}{{{t_2}}}} \right)\]
time taken by the person to reach the top of the tower with this speed is,
\[ \Rightarrow t = \dfrac{d}{v}\]
\[ \Rightarrow t = \dfrac{d}{{d\left( {\dfrac{1}{{{t_1}}} + \dfrac{1}{{{t_2}}}} \right)}}\]
Rearrange the above equation as follows,
\[ \Rightarrow t = \dfrac{1}{{\dfrac{{{t_1} + {t_2}}}{{{t_1}{t_2}}}}}\]
\[ \Rightarrow t = \dfrac{{{t_1}{t_2}}}{{{t_1} + {t_2}}}\]

So, the correct answer is option (D).

Note: In this question, the distance to be traversed is the same for both cases. Therefore, do not take different distances for the distance travelled by the person through escalator and the distance travelled by the person by walking.