Initially a body of mass 10kg is moving along x-axis with velocity `10 sqrt(3)` m/s. It collides with another body of mass 20kg and comes to rest. The 20kg mass object disintegrates in 2 parts each of mass 10kg. One part moves along y-axis with velocity 10m/s and another at 30 degrees with x-axis. Evaluate velocity of object which moves at angle 30 degrees with x-axis

Answer

To solve the problem step by step, we will apply the principle of conservation of momentum.

### Step 1: Understand the initial conditions
Initially, we have:
- Mass of the first body, \( m_1 = 10 \, \text{kg} \)
- Initial velocity of the first body, \( v_1 = 10\sqrt{3} \, \text{m/s} \) (along the x-axis)
- Mass of the second body, \( m_2 = 20 \, \text{kg} \) (initially at rest)

### Step 2: Calculate the initial momentum
The initial momentum of the system in the x-direction is given by:
\[ p_{initial} = m_1 v_1 + m_2 \cdot 0 = 10 \cdot 10\sqrt{3} + 0 = 100\sqrt{3} \, \text{kg m/s} \]
The initial momentum in the y-direction is zero since both bodies are moving along the x-axis.

### Step 3: Analyze the collision
After the collision, the first body comes to rest, and the second body (20 kg) disintegrates into two parts of 10 kg each:
1. One part moves along the y-axis with a velocity of \( 10 \, \text{m/s} \) (let's call this body A).
2. The other part moves at an angle of \( 30^\circ \) with the x-axis (let's call this body B).

### Step 4: Write down the final momentum equations
For the y-direction:
- The momentum of body A (moving upwards) is \( p_A = 10 \cdot 10 = 100 \, \text{kg m/s} \).
- The momentum of body B (moving downwards) is \( p_B = 10 v' \sin(30^\circ) = 10 v' \cdot \frac{1}{2} = 5 v' \).

Since the initial momentum in the y-direction was zero, the final momentum in the y-direction must also be zero:
\[ p_A + p_B = 0 \implies 100 - 5v' = 0 \]

### Step 5: Solve for \( v' \)
Rearranging the equation gives:
\[ 5v' = 100 \implies v' = \frac{100}{5} = 20 \, \text{m/s} \]

### Step 6: Conclusion
The velocity of the object that moves at an angle of \( 30^\circ \) with the x-axis is \( 20 \, \text{m/s} \).

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