A body of mass `M_(1)` collides elastically with another mass `M_(2)` at rest. There is maximum transfer of energy when :

Options

A. `M_(1) gt M_(2)`
B. `M_(1) lt M_(2)`
C. `M_(1) = M_(2)`
D. same for all values of `M_(1)` and `M_(2)`

Answer

To solve the problem of maximum energy transfer during an elastic collision between two bodies of masses \( M_1 \) and \( M_2 \), we can analyze the situation step by step.

### Step 1: Understand Elastic Collision
In an 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 total kinetic energy before the collision is equal to the total kinetic energy after the collision.

### Step 2: Set Up the Problem
Let:
- Mass \( M_1 \) be moving with an initial velocity \( v \).
- Mass \( M_2 \) be at rest (initial velocity = 0).

### Step 3: Apply Conservation of Momentum
The conservation of momentum before and after the collision can be expressed as:
\[ M_1 v + M_2 \cdot 0 = M_1 v_1' + M_2 v_2' \]
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 can be expressed as:
\[ \frac{1}{2} M_1 v^2 + 0 = \frac{1}{2} M_1 (v_1')^2 + \frac{1}{2} M_2 (v_2')^2 \]

### Step 5: Analyze Maximum Energy Transfer
For maximum energy transfer from \( M_1 \) to \( M_2 \), we want the velocity of \( M_2 \) after the collision to be as high as possible. The maximum transfer of energy occurs when the entire kinetic energy of \( M_1 \) is transferred to \( M_2 \). This happens when the masses are equal:
\[ M_1 = M_2 \]

### Conclusion
Thus, the condition for maximum transfer of energy during an elastic collision is when the masses are equal:
\[ M_1 = M_2 \]

### Final Answer
The correct option is that maximum transfer of energy occurs when \( M_1 \) is equal to \( M_2 \).

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