1 kg body explodes into three fragements. The ratio of their masses is `1 : 1: 3`. The fragments of same mass move perpendicular to each other with speed 30 m/s. The speed of heavier part is

Options

A. `(10)/(sqrt(2))` m/s
B. `10sqrt(2)` m/s
C. `20sqrt(2)` m/s
D. `30sqrt(2)` m/s

Explanation

To solve the problem step by step, we will follow the principles of conservation of momentum and the given mass ratios of the fragments.

### Step 1: Determine the Masses of the Fragments
The total mass of the body is given as 1 kg, and the ratio of the masses of the fragments is 1:1:3.

Let the masses of the fragments be:
- \( m_1 = x \)
- \( m_2 = x \)
- \( m_3 = 3x \)

Since the total mass is 1 kg, we have:
\[ m_1 + m_2 + m_3 = 1 \implies x + x + 3x = 1 \implies 5x = 1 \implies x = \frac{1}{5} \text{ kg} \]

Thus, the masses are:
- \( m_1 = \frac{1}{5} \text{ kg} = 0.2 \text{ kg} \)
- \( m_2 = \frac{1}{5} \text{ kg} = 0.2 \text{ kg} \)
- \( m_3 = \frac{3}{5} \text{ kg} = 0.6 \text{ kg} \)

### Step 2: Determine the Momentum of the First Two Fragments
The first two fragments (masses \( m_1 \) and \( m_2 \)) move perpendicular to each other with a speed of 30 m/s.

The momentum \( p \) of an object is given by the product of its mass and velocity:
\[ p_1 = m_1 \cdot v = 0.2 \cdot 30 = 6 \text{ kg m/s} \]
\[ p_2 = m_2 \cdot v = 0.2 \cdot 30 = 6 \text{ kg m/s} \]

### Step 3: Calculate the Resultant Momentum
Since \( p_1 \) and \( p_2 \) are perpendicular, we can find the resultant momentum \( p_3 \) of the third fragment using the Pythagorean theorem:
\[ p_3 = \sqrt{p_1^2 + p_2^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \text{ kg m/s} \]

### Step 4: Find the Speed of the Heavier Fragment
The momentum of the third fragment can also be expressed as:
\[ p_3 = m_3 \cdot v_3 \]
Where \( v_3 \) is the speed of the heavier fragment. We can rearrange this to find \( v_3 \):
\[ v_3 = \frac{p_3}{m_3} = \frac{6\sqrt{2}}{0.6} \]
Calculating this gives:
\[ v_3 = \frac{6\sqrt{2}}{0.6} = 10\sqrt{2} \text{ m/s} \]

### Final Answer
The speed of the heavier part is \( 10\sqrt{2} \text{ m/s} \).

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