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-by-Step Solution:

1. **Understanding the Problem**:
We are dealing with a body moving with uniform acceleration along a straight line. This means that the acceleration (a) is constant.

2. **Using the Relationship Between Acceleration, Velocity, and Position**:
We know that acceleration can be expressed in two ways:
- \( a = \frac{dv}{dt} \) (acceleration as the rate of change of velocity with respect to time)
- \( a = v \frac{dv}{dx} \) (acceleration as the product of velocity and the rate of change of velocity with respect to position)

3. **Setting Up the Equation**:
Since the acceleration is constant, we can set:
\[ a = v \frac{dv}{dx} = c \]
where \( c \) is a constant representing the uniform acceleration.

4. **Rearranging the Equation**:
Rearranging gives us:
\[ v \, dv = c \, dx \]

5. **Integrating Both Sides**:
We integrate both sides:
\[ \int v \, dv = \int c \, dx \]
This results in:
\[ \frac{v^2}{2} = cx + C \]
where \( C \) is the integration constant.

6. **Determining the Constant**:
To find the constant \( C \), we can use initial conditions. If at \( x = 0 \), \( v = 0 \):
\[ \frac{0^2}{2} = c(0) + C \implies C = 0 \]
Thus, the equation simplifies to:
\[ \frac{v^2}{2} = cx \]
or
\[ v^2 = 2cx \]

7. **Final Form of the Equation**:
Rearranging gives us:
\[ v^2 = 2cx \]
This equation shows that the relationship between \( v^2 \) and \( x \) is linear, which is characteristic of a parabolic graph.

8. **Identifying the Graph**:
The equation \( v^2 = 2cx \) resembles the standard form of a parabola \( y = kx \) where \( y \) is \( v^2 \) and \( x \) is the position \( x \). Therefore, the graph of \( v \) versus \( x \) is a parabola opening to the right.

### Conclusion:
The variation of velocity (v) with position (x) for a body moving with uniform acceleration is best represented by a parabolic curve.