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Step-by-Step Solution
Step 1: Understand the Problem
A body of mass $4m$ is initially at rest. It explodes into three pieces: two pieces each of mass $m$ moving perpendicular to each other with speed $v$, and a third piece of mass $2m$. We need to find the total kinetic energy released by this explosion.
Step 2: Apply Conservation of Momentum
Before the explosion, the total momentum of the system is zero because the body is at rest. After the explosion, the vector sum of the momenta of the three pieces must remain zero.
The two pieces of mass $m$ each move perpendicular to each other, so their momenta can be represented as:
Momentum of first piece: $m \cdot v$ along the x-axis (for instance).
Momentum of second piece: $m \cdot v$ along the y-axis (perpendicular to the first one).
The third piece of mass $2m$ must move in such a way that its momentum balances the vector sum of these two perpendicular components. The resultant momentum of the first two pieces is:
$$
\sqrt{(m\,v)^2 + (m\,v)^2} = m v \sqrt{2}.
$$
Hence, to conserve momentum, the momentum of the third piece (mass $2m$) should be equal in magnitude and opposite in direction to this resultant, so:
$$
2m \, v_1 = m v \sqrt{2}.
$$
Therefore,
$$
v_1 = \frac{m v \sqrt{2}}{2m} = \frac{v}{\sqrt{2}}.
$$
Step 3: Compute the Total Kinetic Energy
The kinetic energy (KE) of each piece is given by $\tfrac{1}{2} m_i v_i^2$, where $m_i$ and $v_i$ are the mass and speed of that piece, respectively. Summing for all three pieces:
Kinetic energy of first piece (mass $m$, speed $v$):
$$
\frac{1}{2} m v^2.
$$
Kinetic energy of second piece (mass $m$, speed $v$):
$$
\frac{1}{2} m v^2.
$$
Kinetic energy of third piece (mass $2m$, speed $v_1 = \frac{v}{\sqrt{2}}$):
$$
\frac{1}{2} \cdot 2m \cdot \left(\frac{v}{\sqrt{2}}\right)^2 \;=\; m \cdot \frac{v^2}{2}.
$$
Adding these contributions:
$$
\frac{1}{2} m v^2 + \frac{1}{2} m v^2 + m \cdot \frac{v^2}{2}
= m v^2 + \frac{m v^2}{2}
= \frac{3}{2} m v^2.
$$
Step 4: State the Final Answer
The total kinetic energy released in the explosion is
$$
\frac{3}{2}\, m\, v^2.
$$
