A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and requires 1 s. Plot the x-t graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit 13 m away from the start.

Options

A. 21s
B. 29s
C. 31s
D. 37 s

Answer

To solve the problem step by step, we will analyze the motion of the drunkard, plot the x-t graph, and determine how long it takes for him to fall into the pit located 13 meters away.

### Step 1: Understand the Motion
The drunkard takes 5 steps forward and then 3 steps backward. Each step is 1 meter long and takes 1 second. Therefore, in one complete cycle (5 steps forward and 3 steps backward), he moves:
- Forward: 5 meters
- Backward: 3 meters
- Net movement in one cycle = 5 - 3 = 2 meters

The total time taken for one complete cycle is:
- Time for 5 steps forward = 5 seconds
- Time for 3 steps backward = 3 seconds
- Total time for one cycle = 5 + 3 = 8 seconds

### Step 2: Calculate the Position After Each Cycle
We can calculate the position of the drunkard after each cycle:
1. After 1 cycle (8 seconds): 2 meters
2. After 2 cycles (16 seconds): 4 meters
3. After 3 cycles (24 seconds): 6 meters
4. After 4 cycles (32 seconds): 8 meters
5. After 5 cycles (40 seconds): 10 meters
6. After 6 cycles (48 seconds): 12 meters

### Step 3: Determine the Next Steps
After 6 cycles, the drunkard is at 12 meters. He will take 5 steps forward (to 17 meters) and then 3 steps backward (to 14 meters). Since the pit is at 13 meters, he will fall into the pit during this phase.

### Step 4: Calculate Time to Reach the Pit
- At 48 seconds, he is at 12 meters.
- At 49 seconds, he takes 1 step forward to 13 meters and falls into the pit.

Thus, the total time taken to fall into the pit is:
- 48 seconds (after 6 cycles) + 1 second = 49 seconds

### Step 5: Plot the x-t Graph
1. On the x-axis, we plot the position (in meters).
2. On the y-axis, we plot the time (in seconds).
3. The graph will consist of linear segments:
- From (0, 0) to (5, 5) for the first 5 steps forward.
- From (5, 5) to (2, 8) for the 3 steps backward.
- Repeat this pattern until reaching the pit.

### Final Answer
The drunkard takes **49 seconds** to fall into the pit located 13 meters away from the start.