A particle is projected with initial speed u at an angle `theta` above the horizontal . Let A be the point of projection , B the point where velocity makes an angle `theta //2` above the horizontal and C the highest point of the trajectory .
Radius of curvature of the trajectory at point A is

Options

A. `(u^(2))/(g cos theta)`
B. `(2u^(2))/(g cos theta)`
C. `(u^(2))/(2 g cos theta)`
D. `(u^(2) cos theta)/(g)`

Explanation

To find the radius of curvature of the trajectory at point A, we can follow these steps:

### Step 1: Understand the motion of the projectile
The particle is projected with an initial speed \( u \) at an angle \( \theta \) above the horizontal. The trajectory of the projectile is a parabola.

### Step 2: Identify the forces acting on the particle
At point A (the point of projection), the only forces acting on the particle are the gravitational force \( mg \) acting downwards and the initial velocity \( u \) at an angle \( \theta \).

### Step 3: Determine the components of the initial velocity
The initial velocity \( u \) can be resolved into two components:
- Horizontal component: \( u_x = u \cos(\theta) \)
- Vertical component: \( u_y = u \sin(\theta) \)

### Step 4: Calculate the acceleration
The only acceleration acting on the particle is due to gravity, which is \( g \) acting downwards. Thus, the acceleration in the horizontal direction is \( 0 \) and in the vertical direction is \( -g \).

### Step 5: Find the radius of curvature formula
The radius of curvature \( R \) at any point in a projectile motion can be given by the formula:
\[ R = \frac{(v^2)}{g \cos(\theta)} \]
where \( v \) is the speed of the particle at that point.

### Step 6: Calculate the speed at point A
At point A, the speed \( v \) is equal to the initial speed \( u \) since the particle has just been projected:
\[ v = u \]

### Step 7: Substitute values into the radius of curvature formula
Now substituting \( v = u \) into the radius of curvature formula:
\[ R = \frac{u^2}{g \cos(\theta)} \]

### Final Answer
Thus, the radius of curvature of the trajectory at point A is:
\[ R = \frac{u^2}{g \cos(\theta)} \]

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