All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pic

Q: Why does the velocity-position graph look like a sideways parabola?

From the equation $v^2 = u^2 - 2ax$, we can see that $x$ is proportional to $-v^2$. In a standard $x-y$ plane, $x = -y^2$ is a parabola opening to the left.

Q: Why can't the distance-time graph be a straight line?

A straight line on a distance-time graph implies a constant slope, which means constant speed. Since our particle has a constant negative acceleration, its speed is changing, so the graph must be curved.

Q: What happens to the distance when the velocity becomes negative?

Distance is the total path length covered, so it always increases. Even when velocity is negative (moving backwards), the distance continues to grow. The distance-time graph will never have a negative slope.

Q: How do we know the position-time graph is a downward parabola?

The position equation is $x = ut - \frac{1}{2}at^2$. The negative sign in front of the $t^2$ term indicates that the parabola opens downwards.

Q: Can displacement decrease while distance increases?

Yes! If a particle turns around and moves back towards its starting point, its displacement decreases, but the total distance it has traveled continues to increase.

Question

All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pick it up.

Options:

(A) Velocity vs Position Graph

(B) Distance vs Time Graph

(C) Position vs Time Graph

(D) Velocity vs Time Graph

Sigma Percentile · Accepted Answer

### The Art of Decoding Motion Graphs

Imagine you are a detective, and your suspects are four different graphs. They all claim to describe the exact same motion of a particle. However, one of them is lying. Our mission is to cross-examine each graph, find the underlying physical truth, and catch the imposter!

### The Baseline: Velocity-Time Graph

When dealing with multiple graphs, it is always a great strategy to start with the simplest one. Let's look at the **Velocity vs Time** graph (Graph D).

It shows a straight line with a negative slope. What does this tell us physically? The slope of a velocity-time graph gives us the acceleration. Since the slope is constant and negative, we can confidently say that the particle is moving with a **constant negative acceleration**.

Mathematically, we can write this as:
$$v = u - at$$

This simple equation will be our lie detector test for the other graphs.

### The Consequence: Position-Time Graph

If the acceleration is constant and negative, how should the position change over time? To find out, we integrate our velocity equation:
$$x = \int v \, dt = ut - \frac{1}{2}at^2$$

This is a quadratic equation in time ($t$). In coordinate geometry, an equation of the form $y = -ax^2 + bx$ represents a **parabola opening downwards**.

Now, let's look at Graph C, which plots Position vs Time. It perfectly displays a downward-opening parabola! This means Graph C is telling the truth and is completely consistent with Graph D.

### The Spatial View: Velocity-Position Graph

What if we want to relate velocity directly to position, bypassing time completely? We use the third equation of motion:
$$v^2 = u^2 - 2ax$$

Let's rearrange this to see how position ($x$) depends on velocity ($v$):
$$x = \frac{u^2 - v^2}{2a}$$

If we plot $v$ on the y-axis and $x$ on the x-axis, this equation represents a parabola that opens towards the left (since the coefficient of $v^2$ is negative).

Looking at Graph A, we see exactly this shape—a curve that intersects the x-axis and opens leftwards. Graph A passes the test!

### The Imposter: Distance-Time Graph

Finally, we arrive at Graph B, which plots Distance vs Time. It shows a straight line starting from the origin.

A straight line on a distance-time graph means the slope is constant. Since the slope of distance-time is speed, this graph claims the particle is moving with a **constant speed**.

But wait! We already established from Graph D that the velocity is changing (decreasing linearly). If the velocity is changing, the speed cannot be constant. Therefore, the distance-time graph must be a curve, not a straight line.

Graph B has contradicted our fundamental finding. We have found our imposter!

Animated Solution for Physics - Kinematics: All the graphs below are intended to represent the same motion. One of them does it incorrectly. Pick it up.

Select Answer:

Visualized Solution

$v - t Graph Analysis$

$x - t Graph Analysis$

$v - x Graph Analysis$

$Distance-Time Graph Analysis$

The Sigma Insight: Motion Graphs

The Art of Decoding Motion Graphs

The Baseline

The Consequence

The Spatial View

The Imposter

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