In perfectly elastic collision between two masses `m_(1)and m_(2)` in one dimension energy transfer is a maximum, when

Options

A. `m_(1)=2m_(2)`
B. `m_(1) lt ltm_(2)`
C. `m_(1) gt gt m_(2)`
D. `m_(1)=m_(2)`

Answer

To solve the problem regarding the maximum energy transfer in a perfectly elastic collision between two masses \( m_1 \) and \( m_2 \) in one dimension, we can follow these steps:

### Step 1: Understand the Concept of Elastic Collision
In a perfectly elastic collision, both momentum and kinetic energy are conserved. This means that the total momentum before the collision is equal to the total momentum after the collision, and the same applies to kinetic energy.

### Step 2: Write Down the Conservation Equations
For two masses \( m_1 \) and \( m_2 \) colliding elastically, the conservation of momentum and kinetic energy can be expressed as:
1. **Conservation of Momentum:**
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
where \( u_1 \) and \( u_2 \) are the initial velocities, and \( v_1 \) and \( v_2 \) are the final velocities of masses \( m_1 \) and \( m_2 \) respectively.

2. **Conservation of Kinetic Energy:**
\[ \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]

### Step 3: Analyze Energy Transfer
The energy transfer during the collision can be analyzed by looking at the change in kinetic energy of the masses. The maximum energy transfer occurs when one mass transfers its kinetic energy to the other mass effectively.

### Step 4: Determine Conditions for Maximum Energy Transfer
To find the condition for maximum energy transfer, we can analyze the case when the two masses are equal:
- If \( m_1 = m_2 \), then the velocities after the collision can be derived from the conservation equations.

### Step 5: Conclusion
The energy transfer is maximum when the masses are equal, i.e., \( m_1 = m_2 \). This is because, in this scenario, the velocities after the collision will be such that the kinetic energy is transferred most efficiently from one mass to the other.

### Final Answer
The energy transfer is maximum when \( m_1 = m_2 \).

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