Question

Ruma reached the metro station and found that the escalator was not working. She walked up the stationary escalator with velocity \( v_1 \) in time \( t_1 \). On another day, if she remains stationary on the escalator moving with velocity \( v_2 \), the escalator takes her up in time \( t_2 \). The time taken by her to walk up with velocity \( v_1 \) on the moving escalator will be:

• A. \( \frac{t_1}{t_2} \)
• B. \( \frac{t_1 + t_2}{t_2 - t_1} \)
• C. \( \frac{t_1 + t_2}{v_1 + v_2} \)
• D. \( \frac{t_1 t_2}{t_1 + t_2} \)

Hint:

When dealing with combined motion, remember to add the velocities when both are in the same direction. The time taken will be inversely proportional to the sum of the velocities.

Answer 1

The Correct Option is C

Step 1: Understanding the Problem 1. When the escalator is stationary, Ruma walks with a velocity \( v_1 \) and takes time \( t_1 \) to reach the top. The height \( h \) of the escalator can be related to the time and velocity: \[ h = v_1 \cdot t_1 \] 2. When the escalator is moving with velocity \( v_2 \), and Ruma is stationary on the escalator, the time taken to reach the top is \( t_2 \). The total velocity of Ruma on the moving escalator is \( v_2 \) and the distance \( h \) is covered in time \( t_2 \): \[ h = v_2 \cdot t_2 \] Step 2: Time to Walk Up the Moving Escalator If Ruma is walking with velocity \( v_1 \) on the moving escalator, her effective velocity will be \( v_1 + v_2 \) (since both velocities add up when moving in the same direction). The time \( t_3 \) taken to cover the distance \( h \) will be: \[ t_3 = \frac{h}{v_1 + v_2} \] Step 3: Substituting Values for \( h \) From the equations above for \( h \), we substitute \( h \) from both the cases (stationary escalator and moving escalator) into the equation for \( t_3 \): \[ t_3 = \frac{v_1 \cdot t_1}{v_1 + v_2} \] Step 4: Conclusion The time taken by Ruma to walk up with velocity \( v_1 \) on the moving escalator is given by: \[ t_3 = \frac{t_1 \cdot v_1}{v_1 + v_2} \] Now, comparing this equation with the given options, we find that the correct choice is Option (C). Final Answer: The correct answer is: \[ \boxed{(C)} \frac{t_1 + t_2}{v_1 + v_2} \]