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 1: Understand 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.

### Step 2: Identify Constants
Since the angle \( \theta \) is fixed, \( \sin(2\theta) \) becomes a constant. Therefore, we can denote:
\[ k = \frac{\sin(2\theta)}{g} \]
This allows us to rewrite the range formula as:
\[ R = k u^2 \]

### Step 3: Analyze the Relationship
From the equation \( R = k u^2 \), we can see that the range \( R \) is directly proportional to the square of the initial velocity \( u \). This indicates that as \( u \) increases, \( R \) will increase quadratically.

### Step 4: Graphical Representation
The equation \( R = k u^2 \) represents a parabolic relationship, where \( R \) is plotted on the y-axis and \( u \) on the x-axis. Since \( k \) is a positive constant (as both \( \sin(2\theta) \) and \( g \) are positive), the graph will be an upward-opening parabola.

### Step 5: Conclusion
Thus, the correct graph that represents the variation of range \( R \) versus initial velocity \( u \) is an upward parabola. Among the options provided, we select the graph that shows this upward parabolic shape.

### Final Answer
The correct option is **D**, which shows an upward parabola.

---