A particle is projected from ground at an angle `theta` with horizontal with speed u. The ratio of radius of curvature of its trajectory at point of projection to radius of curvature at maximum height is -

Options

A. `(1)/(sin^(2)thetacostheta)`
B. `cos^(2)theta`
C. `(1)/(sin^(3)theta)`
D. `(1)/(cos^(3)theta)`

Answer

To solve the problem of finding the ratio of the radius of curvature of the trajectory at the point of projection to the radius of curvature at the maximum height, we can follow these steps:

### Step 1: Understand the Concept of Radius of Curvature
The radius of curvature (R) of a trajectory at any point can be calculated using the formula:
\[ R = \frac{v^2}{a_{\perp}} \]
where \( v \) is the velocity of the particle at that point, and \( a_{\perp} \) is the component of acceleration perpendicular to the velocity.

### Step 2: Calculate Radius of Curvature at the Point of Projection (R1)
At the point of projection:
- The initial velocity \( u \) is at an angle \( \theta \) with the horizontal.
- The vertical component of acceleration is \( g \) (downward).
- The horizontal component of the velocity is \( u \cos \theta \).
- The vertical component of the velocity is \( u \sin \theta \).

At the point of projection, the angle between the velocity vector and the vertical acceleration vector is \( 90 - \theta \). Therefore, the perpendicular component of acceleration is:
\[ a_{\perp} = g \cos \theta \]

Now, we can find the radius of curvature at the point of projection:
\[ R_1 = \frac{u^2}{g \cos \theta} \]

### Step 3: Calculate Radius of Curvature at Maximum Height (R2)
At maximum height:
- The vertical component of velocity becomes 0, and only the horizontal component remains, which is \( u \cos \theta \).
- The acceleration acting on the particle is still \( g \) (downward).

Here, the radius of curvature can be calculated as:
\[ R_2 = \frac{(u \cos \theta)^2}{g} = \frac{u^2 \cos^2 \theta}{g} \]

### Step 4: Find the Ratio of R1 to R2
Now we can find the ratio of the radius of curvature at the point of projection to that at maximum height:
\[ \frac{R_1}{R_2} = \frac{\frac{u^2}{g \cos \theta}}{\frac{u^2 \cos^2 \theta}{g}} \]

This simplifies to:
\[ \frac{R_1}{R_2} = \frac{1}{\cos^3 \theta} \]

### Final Answer
Thus, the ratio of the radius of curvature of the trajectory at the point of projection to the radius of curvature at maximum height is:
\[ \frac{R_1}{R_2} = \frac{1}{\cos^3 \theta} \]