Animated Solution for Physics - Kinematics: A particle starts from origin $O$ from rest and moves with a uniform acceleration along the positive X-axis. Identify all figures that correctly represent the motion qualitatively. ($a=\text{acceleration}$, $v=\text{velocity}$, $x=\text{displacement}$, $t=\text{time}$)

Physics is not just about crunching numbers; it is about visualizing reality. When a particle moves, it leaves behind a mathematical footprint. By plotting its acceleration, velocity, and displacement against time, we can read its entire history at a single glance.

In this problem, we are tasked with identifying the correct graphical representations of a particle undergoing a very specific type of motion. Let's break down the physical situation and translate it into the language of geometry.

The problem statement gives us three crucial pieces of information: 1. "Starts from origin $O$": This means the initial displacement is zero. Mathematically, at $t=0$, $x_0=0$. 2. "Starts from rest": The particle is initially stationary. Therefore, its initial velocity is zero. At $t=0$, $u=0$. 3. "Uniform acceleration along the positive X-axis": The acceleration is constant and positive. Let's call this constant value $a$, where $a>0$.

These three conditions are the seeds from which all our graphs will grow.

A Steady Pulse

Let's start with the simplest parameter: acceleration.

We established that the acceleration $a$ is uniform and positive. In the realm of graphs, a constant value over time is represented by a horizontal straight line. Because the acceleration is positive, this line must lie above the time axis.

Looking at the given options, Graph (A) depicts exactly this scenario. The horizontal line indicates that no matter how much time passes, the acceleration remains locked at a steady, positive value. Thus, Graph (A) is a correct representation.

Building Momentum

Next, we move to velocity. How does velocity change when acceleration is constant? We can find out using the first equation of motion:

We know that the particle starts from rest, so we substitute $u=0$ into our equation:

This equation, $v=at$, is a linear equation of the form $y=mx$, where the slope $m$ is the acceleration $a$. Since $a$ is positive, the graph must be a straight line passing through the origin with a positive slope.

As time increases, the velocity increases linearly. Graph (B) perfectly illustrates this linear relationship, starting right from the origin $(0,0)$. Therefore, Graph (B) is also a correct representation.

The Parabolic Arc

Finally, we must determine how the particle's displacement changes over time. For this, we rely on the second equation of motion:

Again, we substitute our initial condition $u=0$:

This is where things get interesting. The displacement $x$ is not proportional to time $t$, but rather to the square of time $t^2$.

In coordinate geometry, an equation of the form $y=k{x}^2$ (where $k$ is a positive constant) represents a parabola that opens upwards and has its vertex at the origin.

Let's evaluate the remaining graphs. Graph (C) shows a straight line for displacement. A linear displacement-time graph implies a constant velocity, which contradicts our premise of a constantly accelerating particle. Therefore, Graph (C) is incorrect.

Graph (D), however, displays a beautiful upward-opening parabolic curve starting from the origin. This perfectly matches our derived equation $x=\frac{1}{2}a{t}^2$. As time ticks on, the particle covers increasingly larger distances in each subsequent second, which is the hallmark of accelerated motion. Thus, Graph (D) is correct.

By systematically applying the equations of kinematics to our initial conditions, we have successfully decoded the motion of the particle.

- The constant acceleration gives us the horizontal line in Graph (A). - The linearly increasing velocity gives us the straight line through the origin in Graph (B). - The quadratically increasing displacement gives us the parabola in Graph (D).

Therefore, the figures that correctly represent the motion qualitatively are (A), (B), and (D). This makes option (d) the correct answer.

Always remember, equations and graphs are two sides of the same coin. Mastering the translation between them is a superpower in physics!