A stone is thrown horizontally with a velocity of `10m//sec`. Find the radius of curvature of it’s trajectory at the end of `3 s` after motion began. `(g=10m//s^(2))`

Options

A. `10sqrt(10)m`
B. `100sqrt(10)m`
C. `sqrt(10)m`
D. `100m`

Explanation

To find the radius of curvature of the stone's trajectory at the end of 3 seconds after it has been thrown horizontally, we can follow these steps:

### Step 1: Determine the horizontal and vertical components of velocity
The stone is thrown horizontally with an initial velocity \( u_x = 10 \, \text{m/s} \). The vertical component of velocity \( v_y \) after time \( t = 3 \, \text{s} \) can be calculated using the equation of motion for free fall:

\[ v_y = g \cdot t \]

Given \( g = 10 \, \text{m/s}^2 \):

\[ v_y = 10 \cdot 3 = 30 \, \text{m/s} \]

### Step 2: Calculate the resultant velocity
The resultant velocity \( v \) can be found using the Pythagorean theorem, since the horizontal and vertical components are perpendicular to each other:

\[ v = \sqrt{u_x^2 + v_y^2} \]

Substituting the values:

\[ v = \sqrt{(10)^2 + (30)^2} = \sqrt{100 + 900} = \sqrt{1000} = 10\sqrt{10} \, \text{m/s} \]

### Step 3: Determine the angle of the resultant velocity
The angle \( \theta \) of the resultant velocity with respect to the horizontal can be found using the tangent function:

\[ \tan(\theta) = \frac{v_y}{u_x} = \frac{30}{10} = 3 \]

### Step 4: Find the component of acceleration perpendicular to the velocity
The only acceleration acting on the stone is the gravitational acceleration \( g \) acting downward. The component of this acceleration that is perpendicular to the velocity can be calculated using:

\[ A_{\perpendicular} = g \sin(\theta) \]

To find \( \sin(\theta) \), we can use the relationship:

\[ \sin(\theta) = \frac{v_y}{v} = \frac{30}{10\sqrt{10}} = \frac{3}{\sqrt{10}} \]

Now substituting this into the equation for \( A_{\perpendicular} \):

\[ A_{\perpendicular} = g \cdot \sin(\theta) = 10 \cdot \frac{3}{\sqrt{10}} = 3\sqrt{10} \, \text{m/s}^2 \]

### Step 5: Calculate the radius of curvature
The radius of curvature \( R \) can be calculated using the formula:

\[ R = \frac{v^2}{A_{\perpendicular}} \]

Substituting the values we have:

\[ R = \frac{(10\sqrt{10})^2}{3\sqrt{10}} = \frac{1000}{3\sqrt{10}} = \frac{100\sqrt{10}}{3} \]

### Final Answer
Thus, the radius of curvature of the trajectory at the end of 3 seconds is:

\[ R = \frac{100\sqrt{10}}{3} \, \text{m} \]