1. Masses of the तीन pieces

Given mass ratio is $1:1:2$. So if total mass is $M$, the three pieces have masses:

$m_1=\frac{M}{4},\quad m_2=\frac{M}{4},\quad m_3=\frac{M}{2}$

2. Use conservation of momentum

Initially, the body is at rest, so total initial momentum is zero. Hence after explosion,

$\vec p_1+\vec p_2+\vec p_3=0$

So,

$\vec p_3=-(\vec p_1+\vec p_2)$

3. Momentum of the two smaller pieces

The two smaller pieces move perpendicular to each other with speeds $30\,\text{m s}^{-1}$ and $40\,\text{m s}^{-1}$.

Their momentum magnitudes are:

$p_1=m_1v_1=\frac{M}{4}\cdot 30=\frac{30M}{4}$

$p_2=m_2v_2=\frac{M}{4}\cdot 40=\frac{40M}{4}$

Since these are perpendicular, resultant momentum magnitude is:

$|\vec p_1+\vec p_2|=\sqrt{p_1^2+p_2^2}$

$=\sqrt{\left(\frac{30M}{4}\right)^2+\left(\frac{40M}{4}\right)^2}$

$=\frac{M}{4}\sqrt{30^2+40^2}$

$=\frac{M}{4}\sqrt{900+1600}$

$=\frac{M}{4}\sqrt{2500}=\frac{50M}{4}$

Therefore,

$p_3=\frac{50M}{4}$

4. Velocity of the third piece

The third piece has mass:

$m_3=\frac{M}{2}$

So its speed is:

$v_3=\frac{p_3}{m_3}$

$v_3=\frac{\frac{50M}{4}}{\frac{M}{2}}$

$v_3=\frac{50}{4}\cdot 2=25\,\text{m s}^{-1}$

5. Option check

• A: $15\,\text{m s}^{-1}$ ❌
• B: $25\,\text{m s}^{-1}$ ✅
• C: $35\,\text{m s}^{-1}$ ❌
• D: $50\,\text{m s}^{-1}$ ❌

Hence the correct answer is B.