A particle is fired from a point on the ground with speed `u` making an angle `theta` with the horizontal.Then

Options

A. the radius of curvature of the projectile at the highest point is `(u^(2)cos^(2)theta)/g`
B. the radius of curvature of the projectile at the highest point is `(u^(2)sin^(2)theta)/g`
C. at the point of projection magnitude of tangential acceleration is `g sin theta`
D. at the point of projection magnitude of tangential acceleration is `g cos theta`

Answer

To solve the problem of finding the radius of curvature of a projectile at its highest point, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components of Velocity**:
- When a particle is fired with speed \( u \) at an angle \( \theta \), it has two components of velocity:
- Horizontal component: \( u_x = u \cos \theta \)
- Vertical component: \( u_y = u \sin \theta \)

2. **Determine the Velocity at the Highest Point**:
- At the highest point of the projectile's trajectory, the vertical component of the velocity becomes zero (\( u_y = 0 \)). Thus, the velocity at the highest point is entirely horizontal:
- \( v = u_x = u \cos \theta \)

3. **Understand the Concept of Radius of Curvature**:
- The radius of curvature \( R \) at any point of a trajectory can be found using the formula:
\[ a = \frac{v^2}{R} \]
- Here, \( a \) is the centripetal acceleration, and \( v \) is the speed of the particle at that point.

4. **Identify the Acceleration at the Highest Point**:
- At the highest point, the only acceleration acting on the projectile is due to gravity, which is \( g \) (acting downwards). This acceleration acts as the centripetal acceleration:
- \( a = g \)

5. **Substitute Values into the Radius of Curvature Formula**:
- Using the values we have:
\[ g = \frac{(u \cos \theta)^2}{R} \]
- Rearranging this equation to solve for \( R \):
\[ R = \frac{(u \cos \theta)^2}{g} \]

6. **Final Expression for Radius of Curvature**:
- Simplifying the expression gives us:
\[ R = \frac{u^2 \cos^2 \theta}{g} \]

### Conclusion:
The radius of curvature of the projectile at the highest point is given by:
\[ R = \frac{u^2 \cos^2 \theta}{g} \]