The Question is from neet-physics , and it was created by the author ABDUL JILANI BASHA

Question

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time ${t_1}.$ On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time ${t_2}.$ The time taken by her to walk up on the moving escalator will be

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
**Primary Concept**: Relative Velocity
**Related Topics**: Speed, Time, Distance, Uniform Motion
**Prerequisites**: Basic understanding of motion, speed, time, and distance relationships.
## SOLUTION PATHWAY
**Given Information**:
* Time taken by Preeti to walk up the stationary escalator: $t_1$
* Time taken by the moving escalator to take Preeti up: $t_2$
* Let the length of the escalator be $d$.
**Method & Steps**:
1. Calculate Preeti's speed when walking: $v_p = \frac{d}{t_1}$
   Reasoning: Speed is distance over time.
   Formula/Rule: $speed = \frac{distance}{time}$
2. Calculate the escalator's speed: $v_e = \frac{d}{t_2}$
   Reasoning: Speed is distance over time.
   Formula/Rule: $speed = \frac{distance}{time}$
3. Calculate Preeti's effective speed when walking on the moving escalator: $v_{eff} = v_p + v_e = \frac{d}{t_1} + \frac{d}{t_2}$
   Reasoning: When walking on a moving escalator, the effective speed is the sum of their individual speeds.
4. Calculate the time taken to walk up the moving escalator: $t = \frac{d}{v_{eff}} = \frac{d}{\frac{d}{t_1} + \frac{d}{t_2}}$
   Reasoning: Time is distance over effective speed.
   Working: $t = \frac{d}{d(\frac{1}{t_1} + \frac{1}{t_2})} = \frac{1}{\frac{t_2 + t_1}{t_1t_2}} = \frac{t_1t_2}{t_1 + t_2}$

5. Final step: $t = \frac{t_1t_2}{t_1+t_2}$
   Result: Time taken is  $\frac{t_1t_2}{t_1+t_2}$
    Verification: The result matches the correct answer D.
## OPTION ANALYSIS
[A] :x: This option suggests subtraction of times which is not correct.
[B] :x: This option suggests averaging of times which is not correct.
[C] :x: This option has subtraction of time in denominator and is incorrect.
[D] ✓: This option correctly calculates the time using the formula derived above.
## LEARNING AIDS
**Common Mistakes**:
1. [Error 1]: Adding the times directly or subtracting them instead of using speeds and distances. [How to avoid]: Understand that speeds need to be added when motion is in the same direction.
2. [Error 2]: Confusing relative speeds with individual speeds. [How to avoid]: Always consider the frame of reference for speed.
**Quick Tips**:
1. [Helpful shortcut]: The formula $\frac{t_1t_2}{t_1+t_2}$ can be remembered for similar problems where two speeds are combined for a fixed distance.
2. [Memory aid]: Remember to add the speeds when motions are in the same direction.

ABDUL JILANI BASHA · Answer

${t_1} - {t_2}$

ABDUL JILANI BASHA · Answer

$\frac{{{t_1} + {t_2}}}{2}$

ABDUL JILANI BASHA · Answer

$\frac{{{t_1}{t_2}}}{{{t_2} - {t_1}}}$

ABDUL JILANI BASHA · Answer

The correct Option is "$\frac{{{t_1}{t_2}}}{{{t_2} + {t_1}}}$"

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