Question

۴۲- با توجه به شکل زیر توضیح دهید کدام یک از نمودارهای مکان - زمان (الف) یا (ب) می تواند نشان دهنده نمودار مکان - زمان یک متحرک باشد.

Explanation:

Analyze graph (a)

$$ \forall t \ge 0, \quad |\{x : (t, x) \in \text{Graph (a)}\}| \le 1 $$

Analyze graph (b)

$$ \exists t > 0 \quad \text{such that} \quad |\{x : (t, x) \in \text{Graph (b)}\}| > 1 $$

Determine physical possibility

$$ \text{Graph (a)} \implies \text{Physically possible (one position per instant)} $$

$$ \text{Graph (b)} \implies \text{Physically impossible (multiple positions at one instant)} $$

Answer:

Graph (a) can represent the position-time graph of a moving object, while Graph (b) cannot.

In Graph (b), drawing a vertical line intersects the curve at more than one point, which physically means the object would have to occupy multiple different positions at the exact same instant in time—an impossibility in classical mechanics. Graph (a) passes the vertical line test, meaning there is exactly one unique position for each instant in time.