What is the radius of curvature of the parabola traced out by the projectile in which a particle is projected with a speed u at an angle `theta` with the horizontal, at a point where the velocity of particle makes an angle `theta//2` with the horizontal.

Explanation

To find the radius of curvature of the parabola traced out by a projectile at a specific point, we can follow these steps:

### Step 1: Understand the problem
We need to find the radius of curvature (RC) of the path of a projectile that is projected with an initial speed \( u \) at an angle \( \theta \) with the horizontal. We are interested in the point where the velocity of the particle makes an angle \( \frac{\theta}{2} \) with the horizontal.

### Step 2: Use the formula for radius of curvature
The formula for the radius of curvature \( R \) in projectile motion is given by:
\[ R = \frac{V^2}{a_{\perp}} \]
where \( V \) is the speed of the projectile at the point of interest, and \( a_{\perp} \) is the component of acceleration perpendicular to the velocity.

### Step 3: Determine the velocity \( V \)
At the point where the velocity makes an angle \( \frac{\theta}{2} \) with the horizontal, we can find the components of the velocity:
- The horizontal component of velocity remains constant:
\[ V_x = u \cos(\theta) \]
- The vertical component of velocity can be found using the kinematic equations. At the point where the angle is \( \frac{\theta}{2} \):
\[ V_y = V \sin\left(\frac{\theta}{2}\right) \]

Using the Pythagorean theorem, the magnitude of the velocity \( V \) can be expressed as:
\[ V = \sqrt{V_x^2 + V_y^2} = \sqrt{(u \cos(\theta))^2 + (V \sin\left(\frac{\theta}{2}\right))^2} \]

### Step 4: Find the perpendicular acceleration \( a_{\perp} \)
The only acceleration acting on the projectile is due to gravity, which acts downward. The component of gravitational acceleration that acts perpendicular to the velocity at the point where the angle is \( \frac{\theta}{2} \) is:
\[ a_{\perp} = g \cos\left(\frac{\theta}{2}\right) \]

### Step 5: Substitute into the radius of curvature formula
Now, substituting \( V \) and \( a_{\perp} \) into the radius of curvature formula:
\[ R = \frac{V^2}{g \cos\left(\frac{\theta}{2}\right)} \]

### Step 6: Find \( V^2 \)
From the earlier steps, we can express \( V^2 \) as:
\[ V^2 = u^2 \cos^2(\theta) + (u \sin(\theta) - gt)^2 \]
However, for simplicity, we can use the relationship derived from the angle:
\[ V^2 = u^2 \cos^2\left(\frac{\theta}{2}\right) \]

### Step 7: Final expression for radius of curvature
Substituting \( V^2 \) into the radius of curvature formula:
\[ R = \frac{u^2 \cos^2\left(\frac{\theta}{2}\right)}{g \cos\left(\frac{\theta}{2}\right)} \]
This simplifies to:
\[ R = \frac{u^2 \cos\left(\frac{\theta}{2}\right)}{g} \]

### Conclusion
Thus, the radius of curvature of the parabola traced out by the projectile at the point where the velocity makes an angle \( \frac{\theta}{2} \) with the horizontal is:
\[ R = \frac{u^2 \cos\left(\frac{\theta}{2}\right)}{g} \]