Two bodies of masses `m_(1)` and `m_2` moving on the same direction with velociities `u_(1)` and `u_2` collide. The velocities after collision are `V_(1)` and `V_2`. If each sphere loses the same amount of kinetic energy, then:

Options

A. `u_(1) + u_(2) + V_(1)-V_(2)=0`
B. `u_(1)-u_(2) + V_(1) + V_(2)=0`
C. `u_(1)+u_(2) +V_(1) + V_(2)=0`
D. `u_(1) + u_(2) -V_(1) -V_(2)=0`

Answer

To solve the problem step by step, we will use the principles of conservation of momentum and the information given about the kinetic energy loss.

### Step 1: Write the conservation of momentum equation
The total momentum before the collision must equal the total momentum after the collision. Thus, we can write:

\[ m_1 u_1 + m_2 u_2 = m_1 V_1 + m_2 V_2 \tag{1} \]

### Step 2: Write the kinetic energy equations
The kinetic energy before the collision is given by:

\[ KE_{\text{initial}} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \]

The kinetic energy after the collision is:

\[ KE_{\text{final}} = \frac{1}{2} m_1 V_1^2 + \frac{1}{2} m_2 V_2^2 \]

### Step 3: Express the kinetic energy loss
Let the amount of kinetic energy lost by each body be \( \Delta KE \). Therefore, we can express the kinetic energy loss for each mass as follows:

For mass \( m_1 \):
\[ \Delta KE_1 = \frac{1}{2} m_1 u_1^2 - \frac{1}{2} m_1 V_1^2 \]

For mass \( m_2 \):
\[ \Delta KE_2 = \frac{1}{2} m_2 u_2^2 - \frac{1}{2} m_2 V_2^2 \]

Since both bodies lose the same amount of kinetic energy, we have:

\[ \Delta KE_1 = \Delta KE_2 \]

### Step 4: Set up the equation for kinetic energy loss
From the above expressions, we can set up the equation:

\[ \frac{1}{2} m_1 u_1^2 - \frac{1}{2} m_1 V_1^2 = \frac{1}{2} m_2 u_2^2 - \frac{1}{2} m_2 V_2^2 \]

### Step 5: Simplify the equation
Cancelling \( \frac{1}{2} \) from both sides gives:

\[ m_1 (u_1^2 - V_1^2) = m_2 (u_2^2 - V_2^2) \]

### Step 6: Factor the differences of squares
Using the identity \( a^2 - b^2 = (a - b)(a + b) \), we can rewrite the equation as:

\[ m_1 (u_1 - V_1)(u_1 + V_1) = m_2 (u_2 - V_2)(u_2 + V_2) \]

### Step 7: Analyze the results
This equation relates the changes in velocities and the masses of the two bodies. It shows that the product of the change in velocity and the sum of the velocities for each mass is proportional to the other mass.

### Final Result
Thus, we have derived a relationship between the masses and velocities before and after the collision under the condition that both bodies lose the same amount of kinetic energy.

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