JEE PYQ: Motion in a Straight Line - Question ID 58d2ee2b36b9 (JEE Main 2025)

In one-dimensional motion, a particle moves along a straight line. Its position \( x(t) \) is a single-valued function of time, meaning at any instant \( t \) there is exactly one position. Consequently:

• The position–time graph must be a single curve without loops or folds (no “overhangs”).
• The velocity–time graph must be a single-valued function of \( t \), because velocity \( v(t) = \frac{dx}{dt} \) is uniquely determined at each instant.
• The acceleration–time graph must likewise be single-valued.

Key relations: $v(t) = \frac{dx}{dt},\qquad a(t) = \frac{dv}{dt}.$

Examine each graph in turn:

1. Graph A (position–time):

   The curve is smooth and never “doubles back” on itself. At each \( t \) there is exactly one \( x \). Hence it can represent one-dimensional motion.

2. Graph B (velocity–time):

   The curve is a single-valued function of \( t \). Although it has sharp corners (discontinuous acceleration), at no instant does the velocity take two values. Therefore it is admissible.

3. Graph C (position–time):

   The curve loops back on itself, producing two different positions at the same time. This violates the single-valued requirement for \( x(t) \). Hence it cannot represent one-dimensional motion.

4. Graph D (acceleration–time):

   The graph is a single-valued step function. Although the acceleration jumps, at each instant there is only one value. Thus it is physically acceptable.

Conclusion: Graphs A, B, and D satisfy the single-valued condition; graph C does not. The correct option is therefore C: A, B and D only.

Students often confuse “smoothness” with “single-valuedness.” A graph can have sharp corners (discontinuous derivatives) yet still represent valid one-dimensional motion, provided it never assigns two positions (or velocities) to the same instant. Conversely, a smooth-looking loop in a position–time graph is always invalid.

Tip: Always check whether the curve is a function of time—if you can draw a vertical line that intersects the curve more than once, the graph cannot describe one-dimensional motion.