A projectile is projected with a velocity u at an angle `theta` with the horizontal. For a fixed `theta`, which of the graphs shown in the following figure shows the variation of range R versus u?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the variation of range \( R \) versus initial velocity \( u \) for a projectile launched at a fixed angle \( \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Range Formula**:
The range \( R \) of a projectile launched with an initial velocity \( u \) at an angle \( \theta \) is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \( g \) is the acceleration due to gravity.

2. **Identifying Fixed Parameters**:
In this scenario, the angle \( \theta \) is fixed. This means that \( \sin(2\theta) \) is a constant value for a given \( \theta \). Therefore, we can denote this constant as \( k = \sin(2\theta) \).

3. **Rewriting the Range Formula**:
Substituting \( k \) into the range formula, we get:
\[ R = \frac{u^2 k}{g} \]
This shows that the range \( R \) is directly proportional to the square of the initial velocity \( u \).

4. **Establishing the Relationship**:
Since \( R \) is proportional to \( u^2 \), we can express this relationship as:
\[ R \propto u^2 \]
This indicates that as \( u \) increases, \( R \) increases with the square of \( u \).

5. **Graphical Representation**:
The relationship \( R \propto u^2 \) suggests that if we plot \( R \) on the y-axis and \( u \) on the x-axis, the graph will be a parabola opening upwards. The general form of the equation for such a graph is:
\[ R = k' u^2 \]
where \( k' \) is a constant.

6. **Identifying the Correct Graph**:
Among the options provided, the graph that represents a quadratic relationship (a parabola) is the one that shows \( R \) increasing as \( u^2 \). Therefore, the correct option is the one that resembles the shape of a parabola.

### Conclusion:
The graph that shows the variation of range \( R \) versus initial velocity \( u \) is a parabola, confirming that \( R \) is proportional to \( u^2 \).