For a body moving with uniform acceleration along straight line, the variation of its velocity (v) with position (x) is best represented by

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Answer

To solve the problem of how the velocity (v) of a body moving with uniform acceleration varies with its position (x), we can follow these steps:

### Step 1: Understand the relationship between acceleration, velocity, and position.
For a body moving with uniform acceleration, we know that acceleration (a) is constant. The relationship between velocity (v), acceleration (a), and position (x) can be derived from the basic kinematic equations.

### Step 2: Use the definition of acceleration.
Acceleration is defined as the rate of change of velocity with respect to time:
\[ a = \frac{dv}{dt} \]

### Step 3: Express acceleration in terms of velocity and position.
Using the chain rule, we can express acceleration in terms of velocity and position:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \frac{dv}{dx} \]
Thus, we can rewrite the acceleration as:
\[ a = v \frac{dv}{dx} \]

### Step 4: Relate acceleration to the change in velocity.
Since acceleration is constant, we can set it equal to a constant value (let's denote it as \( C \)):
\[ v \frac{dv}{dx} = C \]

### Step 5: Rearrange and integrate.
Rearranging gives us:
\[ v \, dv = C \, dx \]
Now, we can integrate both sides. The left side integrates from the initial velocity \( 0 \) to \( v \), and the right side integrates from the initial position \( 0 \) to \( x \):
\[ \int_0^v v \, dv = \int_0^x C \, dx \]

### Step 6: Perform the integration.
The left side becomes:
\[ \frac{v^2}{2} \Big|_0^v = \frac{v^2}{2} \]
The right side becomes:
\[ C x \Big|_0^x = C x \]
Thus, we have:
\[ \frac{v^2}{2} = C x \]

### Step 7: Solve for velocity.
Multiplying both sides by 2 gives:
\[ v^2 = 2Cx \]

### Step 8: Identify the relationship.
This equation \( v^2 = 2Cx \) indicates that the relationship between velocity and position is quadratic. If we rearrange it, we can express it as:
\[ v^2 = 2C x \]
This is similar to the standard form of a parabola \( y^2 = 4Ax \), where \( A \) is a constant.

### Conclusion:
The variation of velocity (v) with position (x) for a body moving with uniform acceleration is best represented by a parabolic curve, specifically \( v^2 \) as a function of \( x \).

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