A 1 kg stationary bomb is exploded in three parts having mass 1 : 1 : 3 respectively. Parts having same mass move in perpendicular direction with velocity `30 ms^(–1)`, then the velocity of bigger part will be : -

Options

A. `10sqrt2 ms^(-1)`
B. `10/sqrt2ms^(-1)`
C. `15sqrt2ms^(-1)`
D. `15/sqrt2ms^(-1)`

Explanation

To solve the problem step by step, we will use the principle of conservation of momentum. Here's the detailed solution:

### Step 1: Understand the Problem
We have a bomb with a total mass of 1 kg that explodes into three parts with mass ratios of 1:1:3. The two smaller parts (each of mass 0.2 kg) move in perpendicular directions with a velocity of 30 m/s. We need to find the velocity of the larger part (mass 0.6 kg).

### Step 2: Determine the Masses
Given the mass ratio of 1:1:3, we can denote the masses as:
- Mass of part 1 (m1) = x
- Mass of part 2 (m2) = x
- Mass of part 3 (m3) = 3x

The total mass is:
\[ m1 + m2 + m3 = x + x + 3x = 5x = 1 \text{ kg} \]
Thus, we find:
\[ x = \frac{1}{5} = 0.2 \text{ kg} \]
So, the masses are:
- m1 = 0.2 kg
- m2 = 0.2 kg
- m3 = 0.6 kg

### Step 3: Set Up the Momentum Conservation Equation
Since the bomb is initially stationary, the initial momentum is zero. After the explosion, the momentum must also equal zero:
\[ 0 = m1 \cdot v1 + m2 \cdot v2 + m3 \cdot v3 \]

Where:
- \( v1 \) and \( v2 \) are the velocities of the two smaller parts,
- \( v3 \) is the velocity of the larger part.

### Step 4: Assign Directions to the Velocities
Assume:
- Part 1 moves in the positive x-direction: \( v1 = 30 \, \text{m/s} \, \hat{i} \)
- Part 2 moves in the positive y-direction: \( v2 = 30 \, \text{m/s} \, \hat{j} \)

### Step 5: Substitute Values into the Momentum Equation
Substituting the values into the momentum equation:
\[ 0 = (0.2 \cdot 30 \hat{i}) + (0.2 \cdot 30 \hat{j}) + (0.6 \cdot v3) \]
This simplifies to:
\[ 0 = 6 \hat{i} + 6 \hat{j} + 0.6 v3 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 0.6 v3 = -6 \hat{i} - 6 \hat{j} \]
Thus:
\[ v3 = \frac{-6 \hat{i} - 6 \hat{j}}{0.6} \]
\[ v3 = -10 \hat{i} - 10 \hat{j} \]

### Step 7: Calculate the Magnitude of the Velocity
The magnitude of \( v3 \) is given by:
\[ |v3| = \sqrt{(-10)^2 + (-10)^2} \]
\[ |v3| = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \, \text{m/s} \]

### Step 8: Conclusion
The velocity of the larger part is \( 10\sqrt{2} \, \text{m/s} \) in the direction opposite to the resultant of the two smaller parts.