A man standing on a moving escalator, completes a certain distance in time `t_(1)`. If the escalator does not move then man covers this distance in time `t_(2)` by walking.
(i) How much time will he take to cover the same distance if he move on the moving escalator in the same direction ?
(ii) How much time will he take to cover the same distance if he move on the moving escalator in the opposite direction ?

Answer

To solve the problem step by step, we need to analyze the situation involving the man, the escalator, and the distances covered.

### Given:
- Distance covered by the man = \( x \)
- Time taken when the man is standing on the moving escalator = \( t_1 \)
- Time taken when the escalator is stationary and the man walks = \( t_2 \)

### Step 1: Determine the speeds
1. **Speed of the escalator (v1)**:
\[ v_1 = \frac{x}{t_1} \]

2. **Speed of the man walking (v2)**:
\[ v_2 = \frac{x}{t_2} \]

### Step 2: Case (i) - Man moving in the same direction as the escalator
When the man walks in the same direction as the escalator, the relative speed (v) is the sum of the speeds:
\[ v = v_1 + v_2 = \frac{x}{t_1} + \frac{x}{t_2} \]

Now, we can express this in terms of time taken (T) to cover the distance \( x \):
\[ T = \frac{x}{v} = \frac{x}{\left(\frac{x}{t_1} + \frac{x}{t_2}\right)} \]

### Step 3: Simplifying the expression
Factor out \( x \):
\[ T = \frac{x}{x \left(\frac{1}{t_1} + \frac{1}{t_2}\right)} = \frac{1}{\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} \]

This can be simplified further:
\[ T = \frac{t_1 \cdot t_2}{t_1 + t_2} \]

### Step 4: Case (ii) - Man moving in the opposite direction to the escalator
When the man walks in the opposite direction to the escalator, the relative speed (v) is the difference of the speeds:
\[ v = v_1 - v_2 = \frac{x}{t_1} - \frac{x}{t_2} \]

Now, we can express this in terms of time taken (T) to cover the distance \( x \):
\[ T = \frac{x}{v} = \frac{x}{\left(\frac{x}{t_1} - \frac{x}{t_2}\right)} \]

### Step 5: Simplifying the expression
Factor out \( x \):
\[ T = \frac{x}{x \left(\frac{1}{t_1} - \frac{1}{t_2}\right)} = \frac{1}{\left(\frac{1}{t_1} - \frac{1}{t_2}\right)} \]

This can be simplified further:
\[ T = \frac{t_1 \cdot t_2}{t_1 - t_2} \]

### Final Answers:
(i) Time taken to cover the distance in the same direction:
\[ T = \frac{t_1 \cdot t_2}{t_1 + t_2} \]

(ii) Time taken to cover the distance in the opposite direction:
\[ T = \frac{t_1 \cdot t_2}{t_1 - t_2} \]