A particle moves along the plane trajectory y(x) with constant speed 'v'. The trajectory has the form of a parabola `y = ax^(2)` where 'a' is a positive constant. Then the radius of curvature of the trajectory at the point x=0 is

Options

A. `(1)/(2a)`
B. `(1)/(4a)`
C. `(1)/(3a)`
D. `(1)/(a)`

Answer

To find the radius of curvature of the trajectory given by the equation \( y = ax^2 \) at the point \( x = 0 \), we can follow these steps:

### Step 1: Understand the Formula for Radius of Curvature
The radius of curvature \( R \) for a curve defined by \( y = f(x) \) can be calculated using the formula:
\[ R = \frac{(1 + (dy/dx)^2)^{3/2}}{d^2y/dx^2} \]

### Step 2: Differentiate the Function
Given the function \( y = ax^2 \):
1. First, we need to find the first derivative \( dy/dx \):
\[ \frac{dy}{dx} = \frac{d}{dx}(ax^2) = 2ax \]

### Step 3: Find the Second Derivative
2. Next, we find the second derivative \( d^2y/dx^2 \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2ax) = 2a \]

### Step 4: Evaluate at \( x = 0 \)
3. Now, we evaluate both derivatives at \( x = 0 \):
- \( \frac{dy}{dx} \) at \( x = 0 \):
\[ \frac{dy}{dx} \bigg|_{x=0} = 2a(0) = 0 \]
- \( \frac{d^2y}{dx^2} \) is constant:
\[ \frac{d^2y}{dx^2} = 2a \]

### Step 5: Substitute into the Radius of Curvature Formula
4. Substitute these values into the radius of curvature formula:
\[ R = \frac{(1 + (0)^2)^{3/2}}{2a} = \frac{1^{3/2}}{2a} = \frac{1}{2a} \]

### Conclusion
Thus, the radius of curvature of the trajectory at the point \( x = 0 \) is:
\[ R = \frac{1}{2a} \]