If two masses `m_(1)` and `m_(2)` collide, the ratio of change in their respective velocities is proportional to

Options

A. `(m_(1))/(m_(2))`
B. `sqrt((m_(1))/(m_(2)))`
C. `(m_(2))/(m_(1))`
D. `sqrt((m_(2))/(m_(1)))`

Explanation

To solve the problem of finding the ratio of change in velocities of two masses \( m_1 \) and \( m_2 \) after a collision, we can follow these steps:

### Step 1: Understand the Initial and Final Velocities
Let:
- \( U_1 \) = initial velocity of mass \( m_1 \)
- \( V_1 \) = final velocity of mass \( m_1 \)
- \( U_2 \) = initial velocity of mass \( m_2 \)
- \( V_2 \) = final velocity of mass \( m_2 \)

### Step 2: Apply the Conservation of Momentum
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. This can be expressed as:
\[ m_1 U_1 + m_2 U_2 = m_1 V_1 + m_2 V_2 \]

### Step 3: Define the Change in Velocities
The change in velocity for each mass can be defined as:
- Change in velocity of \( m_1 \): \( \Delta V_1 = V_1 - U_1 \)
- Change in velocity of \( m_2 \): \( \Delta V_2 = V_2 - U_2 \)

### Step 4: Rearrange the Conservation of Momentum Equation
From the conservation of momentum equation, we can rearrange it to isolate the changes in velocities:
\[ m_1 (U_1 - V_1) = m_2 (V_2 - U_2) \]

### Step 5: Express the Ratio of Changes in Velocities
We can express the ratio of the changes in velocities as:
\[ \frac{U_1 - V_1}{V_2 - U_2} = \frac{m_2}{m_1} \]

### Step 6: Finalize the Ratio
Thus, the ratio of the change in velocities is:
\[ \frac{\Delta V_1}{\Delta V_2} = \frac{m_2}{m_1} \]

### Conclusion
The ratio of the change in velocities of the two masses \( m_1 \) and \( m_2 \) after the collision is given by:
\[ \frac{\Delta V_1}{\Delta V_2} = \frac{m_2}{m_1} \]