Answer the following giving reasons in brief: Is the time variation of position, shown in the figure observed in nature?

Step 1: Analyze the axes and features of the graph.

The graph plots position $x$ on the horizontal axis and time $t$ on the vertical axis. The curve forms an arch starting at $x_1$ and ending at $x_2$ on the position axis.

At both endpoints, the time is zero:

$$t = 0 \quad \text{at} \quad x = x_1 \quad \text{and} \quad x = x_2$$

The peak of the arch represents the maximum time:

$$t = t_{\max} \quad \text{at} \quad x = x_m$$

Step 2: Determine the number of positions at a given instant of time.

Drawing a horizontal line at any constant value of time $t$ (where $0 < t < t_{\max}$) intersects the curve at two distinct positions.

$$x = x_a \quad \text{and} \quad x = x_b \quad (\text{where } x_a \neq x_b)$$

This indicates that at a single instant of time, the particle exists at two different locations simultaneously.

Step 3: Evaluate the physical possibility of the motion.

For any real physical motion, a particle can occupy exactly one position at any given instant. Thus, position must be a single-valued function of time:

$$x = x(t)$$

Since this graph yields two different positions for a single value of time, it represents a physical impossibility.

Answer: No, this time variation of position is not observed in nature because a particle cannot exist at two different positions at the same instant of time.