Show that in a head on elastic collision between two particles, the transference of energy is maximum when their mass ratio is unity.

Explanation

To show that in a head-on elastic collision between two particles, the transference of energy is maximum when their mass ratio is unity, we can follow these steps:

### Step 1: Understand the scenario
Consider two particles (or balls) with masses \( m_1 \) and \( m_2 \). Let \( u_1 \) be the initial velocity of mass \( m_1 \) and \( u_2 \) be the initial velocity of mass \( m_2 \). We assume that \( m_1 \) is moving towards \( m_2 \), which is initially at rest.

### Step 2: Write the equations for kinetic energy
The kinetic energy (KE) of each mass before the collision is given by:
- For mass \( m_1 \):
\[ KE_1 = \frac{1}{2} m_1 u_1^2 \]
- For mass \( m_2 \):
\[ KE_2 = \frac{1}{2} m_2 u_2^2 \]

### Step 3: Apply conservation of momentum
In an elastic collision, both momentum and kinetic energy are conserved. The conservation of momentum before and after the collision can be expressed as:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
where \( v_1 \) and \( v_2 \) are the velocities of \( m_1 \) and \( m_2 \) after the collision.

### Step 4: Apply conservation of kinetic energy
The conservation of kinetic energy in an elastic collision states that:
\[ \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]

### Step 5: Analyze the case when \( m_1 = m_2 \)
Assuming \( m_1 = m_2 = m \), we can simplify the equations. The initial kinetic energy becomes:
\[ KE_{initial} = \frac{1}{2} m u_1^2 + \frac{1}{2} m u_2^2 \]
After the collision, if the velocities are exchanged (which is true for equal masses), we have:
\[ v_1 = u_2 \quad \text{and} \quad v_2 = u_1 \]
Thus, the final kinetic energy becomes:
\[ KE_{final} = \frac{1}{2} m u_2^2 + \frac{1}{2} m u_1^2 \]
This shows that the total kinetic energy is conserved and maximally transferred when the masses are equal.

### Step 6: Conclusion
From the analysis, we can conclude that the maximum transfer of energy occurs when the mass ratio \( \frac{m_1}{m_2} = 1 \). This means that for maximum energy transfer in a head-on elastic collision, the two masses must be equal.