Question

Which of the following curves possibly represent one-dimensional motion of a particle?

• A. A, B and D Only
• B. A, D, and C Only
• C. C, D Only
• D. B, and D Only

Hint:

Analyze each graph to determine if it can represent one-dimensional motion. Consider the physical quantities involved and their relationships.

Answer 1

The Correct Option is A

The objective is to determine which of the four provided curves can plausibly depict the one-dimensional motion of a particle. Each graph must be assessed for its physical validity.

Concept Used:

To ascertain the physical possibility of a graph representing motion, it must align with fundamental kinematic principles.

1. Single-Valued Functions: At any given instant, a particle can only occupy a single position, possess a single velocity, or have a single value for any other state variable. Consequently, any graph plotting a physical quantity against time must represent a single-valued function of time.

2. Properties of Distance: The total distance traversed by a particle is a non-negative scalar quantity that cannot decrease over time. The gradient of a distance-time graph corresponds to the particle's speed, which must also be non-negative.

3. Simple Harmonic Motion (SHM): SHM serves as a common illustration of one-dimensional motion, and its characteristics are frequently visualized through graphs.

• Displacement: \( x = A \sin(\omega t + \phi_0) \)
• Velocity: \( v = A\omega \cos(\omega t + \phi_0) \)
• Phase: The term \( (\omega t + \phi_0) \) represents the phase, which escalates linearly with time.
• The interrelation between velocity and displacement is \( v = \pm \omega \sqrt{A^2 - x^2} \), reformulable as \( \frac{x^2}{A^2} + \frac{v^2}{(A\omega)^2} = 1 \). This equation describes an ellipse in the velocity-displacement plane (a phase space diagram).

Step-by-Step Solution:

Step 1: Analysis of Curve (A) - Phase vs. Time

This graph illustrates the particle's phase as a linear function of time, passing through the origin. This can be described by \( \text{Phase} = kt \), where k is a constant. Such a relationship is valid for oscillatory motions like SHM, where phase is defined as \( \phi(t) = \omega t + \phi_0 \). With \( \phi_0 = 0 \), the phase increases linearly with time. As SHM is a one-dimensional motion, this curve is physically plausible. Thus, (A) is a potential representation.

Step 2: Analysis of Curve (B) - Velocity vs. Displacement

This graph displays a circular correlation between velocity and displacement, indicative of a phase space diagram for an oscillator. For SHM, the relationship is \( \frac{x^2}{A^2} + \frac{v^2}{(A\omega)^2} = 1 \), which forms an ellipse. If \( \omega = 1 \) rad/s and the axes are scaled appropriately, this ellipse becomes a circle. At any given displacement \(x\), the particle can exhibit either a positive velocity (moving away from the origin) or a negative velocity (moving towards the origin), consistent with the graph. Since SHM is one-dimensional, this curve is physically tenable. Therefore, (B) is a potential representation.

Step 3: Analysis of Curve (C) - Velocity vs. Time

This graph exhibits a circular relationship between velocity and time. A vertical line drawn at a specific time t (excluding endpoints) would intersect the circle at two points, implying the particle possesses two distinct velocities (one positive, one negative) simultaneously. This scenario is physically impossible, as a particle's velocity must be uniquely defined at any given moment. Consequently, (C) is not a valid representation.

Step 4: Analysis of Curve (D) - Total Distance vs. Time

This graph plots the total distance traveled against time. Crucially, total distance cannot decrease, and its rate of change (speed) cannot be negative. The depicted graph shows distance increasing, then remaining constant (indicating the particle is momentarily stationary), and then increasing again. The distance never diminishes, and the curve remains a single-valued function of time. This represents a valid physical situation, such as a particle moving at a constant speed, pausing, and then resuming motion. Therefore, (D) is a potential representation.

Final Computation & Result:

Based on the analysis, curves (A), (B), and (D) can plausibly represent one-dimensional particle motion. Curve (C) is physically unachievable.

Considering the provided options:

(1) A, B and D only

(2) A, B and C only

(3) A and B only

(4) A, C and D only

The correct combination of plausible curves is presented in option (1).

The correct answer is (1) A, B and D only.