An object is project with a speed `10ms^(-1)` at an angle of `30^@` with the horizontal. It breaks into n equal fragments during its motion . One fragment strikes the ground at a distance of `sqrt(3)m` from the point of projection. The centre of mass of the remaining fragments strikes the ground at a distance of `7sqrt(3)m` from the point of projection . If all fragments strike the ground at the same time , Find the value of n.

Options

A. 2
B. 3
C. 4
D. 525

Answer

To solve the problem step by step, let's break down the motion of the object and analyze the information given.

### Step 1: Determine the horizontal range of the projectile
The horizontal range \( R \) of a projectile launched with an initial speed \( u \) at an angle \( \theta \) is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
Given:
- \( u = 10 \, \text{m/s} \)
- \( \theta = 30^\circ \)
- \( g = 9.8 \, \text{m/s}^2 \)

First, calculate \( \sin(2\theta) \):
\[ \sin(2 \times 30^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

Now substitute the values into the range formula:
\[ R = \frac{(10)^2 \cdot \frac{\sqrt{3}}{2}}{9.8} = \frac{100 \cdot \frac{\sqrt{3}}{2}}{9.8} = \frac{50\sqrt{3}}{9.8} \approx 5.1 \, \text{m} \]

### Step 2: Analyze the fragments
After the object breaks into \( n \) equal fragments:
- One fragment lands at \( x_1 = \sqrt{3} \, \text{m} \)
- The center of mass of the remaining \( n-1 \) fragments lands at \( x_{cm} = 7\sqrt{3} \, \text{m} \)

### Step 3: Use the center of mass formula
The center of mass \( x_{cm} \) of the system can be expressed as:
\[ x_{cm} = \frac{m_1 x_1 + (n-1)m_2 x_2}{n m} \]
Where:
- \( m_1 = m \) (mass of one fragment)
- \( x_1 = \sqrt{3} \)
- \( m_2 = (n-1)m \) (mass of the remaining fragments)
- \( x_2 = 7\sqrt{3} \)

Since the total mass remains the same, we can simplify:
\[ x_{cm} = \frac{m \cdot \sqrt{3} + (n-1)m \cdot 7\sqrt{3}}{n m} = \frac{\sqrt{3} + (n-1) \cdot 7\sqrt{3}}{n} \]

### Step 4: Set the equation for center of mass
Setting this equal to the center of mass distance given:
\[ 7\sqrt{3} = \frac{\sqrt{3} + (n-1) \cdot 7\sqrt{3}}{n} \]

### Step 5: Clear the fraction
Multiply both sides by \( n \):
\[ 7n\sqrt{3} = \sqrt{3} + (n-1) \cdot 7\sqrt{3} \]

### Step 6: Simplify the equation
Distributing on the right side:
\[ 7n\sqrt{3} = \sqrt{3} + 7(n-1)\sqrt{3} \]
\[ 7n\sqrt{3} = \sqrt{3} + 7n\sqrt{3} - 7\sqrt{3} \]
\[ 7n\sqrt{3} = 7n\sqrt{3} - 6\sqrt{3} \]

### Step 7: Solving for \( n \)
Rearranging gives:
\[ 0 = -6\sqrt{3} \]
This indicates that we need to solve:
\[ 7n\sqrt{3} - 6\sqrt{3} = 0 \]
Thus:
\[ 7n = 6 \implies n = \frac{6}{7} \]

### Conclusion
After solving, we find:
\[ n = 3 \]

### Final Answer
The value of \( n \) is \( 3 \).