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.

Question

A car starts from rest and accelerates uniformly at 3 m/s\textsuperscript{2. What will be its velocity after 5 seconds?}

Answer 1

Given:

Each step length = 1 m
Time for each step = 1 s

Motion pattern:
5 steps forward → 3 steps backward → repeat

Net displacement in one cycle:
= 5 − 3 = 2 m

Time taken for one cycle:
= 5 + 3 = 8 s

Part 1: x–t graph of the motion

The motion is non-uniform and consists of straight-line segments.

Key points for plotting the x–t graph:

Join successive points by straight lines to obtain the x–t graph.

Part 2: Time taken to fall into the pit (13 m away)

(a) Graphical method:

On the x–t graph, draw a horizontal line at x = 13 m.
The time coordinate where this line intersects the motion graph gives the required time.

From the graph, the intersection occurs during a forward motion segment at about:
t ≈ 35 s

(b) Analytical (otherwise) method:

Net displacement per cycle = 2 m
Time per cycle = 8 s

After 4 complete cycles:

Displacement = 4 × 2 = 8 m
Time = 4 × 8 = 32 s

After 32 s, the drunkard starts the next forward motion from x = 8 m.

He needs to reach x = 13 m, so additional distance required:

13 − 8 = 5 m

Since each forward step is 1 m per second:

Additional time required = 5 s

Total time taken:

t = 32 + 5 = 37 s

However, the drunkard falls into the pit exactly at the end of the fifth forward step, so the fall occurs at:

t = 35 s

Final Answer:

The x–t graph is a zig-zag (saw-tooth) curve made of straight-line segments.
The drunkard falls into the pit after 35 seconds.

Solution and Explanation