A body of mass 4m is split into two equal parts by an internal explosion which generates a kinetic energy E. If, after the explosion, the parts move in the same line as before, then what is their relative speed?

Options

A. `sqrt(E//m)`
B. `sqrt(2E//m)`
C. `sqrt(4E//m)`
D. 0

Answer

To solve the problem step by step, we will follow these logical steps:

### Step 1: Understand the Problem
We have a body of mass \(4m\) that splits into two equal parts due to an internal explosion, generating kinetic energy \(E\). We need to find the relative speed of the two parts after the explosion.

### Step 2: Define the Mass of Each Part
Since the body of mass \(4m\) splits into two equal parts, the mass of each part will be:
\[ m_1 = m_2 = 2m \]

### Step 3: Apply Conservation of Momentum
Before the explosion, the body is at rest, so the initial momentum is:
\[ \text{Initial Momentum} = 0 \]
After the explosion, let the velocities of the two parts be \(v_1\) and \(v_2\). By conservation of momentum:
\[ 0 = 2m \cdot v_1 + 2m \cdot (-v_2) \]
This simplifies to:
\[ 0 = 2m v_1 - 2m v_2 \]
Thus, we can conclude:
\[ v_1 = v_2 \]

### Step 4: Determine the Kinetic Energy
The total kinetic energy generated by the explosion is given as \(E\). The kinetic energy of each part can be expressed as:
\[ E = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]
Substituting \(m_1\) and \(m_2\):
\[ E = \frac{1}{2} (2m) v_1^2 + \frac{1}{2} (2m) v_2^2 \]
Since \(v_1 = v_2 = v\), we can write:
\[ E = 2m \left(\frac{1}{2} v^2 + \frac{1}{2} v^2\right) = 2m v^2 \]
This simplifies to:
\[ E = mv^2 \]

### Step 5: Solve for Velocity \(v\)
From the equation \(E = mv^2\), we can solve for \(v\):
\[ v^2 = \frac{E}{m} \]
Taking the square root gives:
\[ v = \sqrt{\frac{E}{m}} \]

### Step 6: Calculate the Relative Speed
The relative speed \(v_r\) between the two parts is given by:
\[ v_r = v_1 - (-v_2) = v_1 + v_2 = v + v = 2v \]
Substituting the value of \(v\):
\[ v_r = 2\sqrt{\frac{E}{m}} \]

### Step 7: Final Expression for Relative Speed
Thus, the final expression for the relative speed of the two parts after the explosion is:
\[ v_r = 2\sqrt{\frac{E}{m}} \]