A ball of mass '2m' moving with velocity `u hat(i)` collides with another ball of mass 'm' moving with velocity `- 2 u hat(i)`. After the collision, mass 2m moves with velocity `(u)/(2) ( hat(i) - hat(j))`. Then the change in kinetic energy of the system ( both balls ) is

Options

A. zero
B. `(3)/(2) m u^(2)`
C. `(3)/(2) m u^(2) ( 1- ( 1)/( 2 sqrt(2)))`
D. `(3)/(2) m u^(2) ( 1+ ( 1)/( 2 sqrt(2)))`

Answer

To find the change in kinetic energy of the system after the collision, we will follow these steps:

### Step 1: Calculate Initial Kinetic Energy
The initial kinetic energy (KE_initial) of the system can be calculated using the formula:
\[ KE = \frac{1}{2} m v^2 \]
For the two balls:
- Ball 1 (mass = 2m, velocity = \(u \hat{i}\)):
\[ KE_1 = \frac{1}{2} (2m) (u^2) = mu^2 \]
- Ball 2 (mass = m, velocity = \(-2u \hat{i}\)):
\[ KE_2 = \frac{1}{2} (m) ((-2u)^2) = \frac{1}{2} m (4u^2) = 2mu^2 \]
Thus, the total initial kinetic energy is:
\[ KE_{\text{initial}} = KE_1 + KE_2 = mu^2 + 2mu^2 = 3mu^2 \]

### Step 2: Calculate Final Kinetic Energy
After the collision, the final velocities are given:
- Ball 1 (mass = 2m, final velocity = \(\frac{u}{2} (\hat{i} - \hat{j})\)):
\[ v_1 = \frac{u}{2} \sqrt{1^2 + 1^2} = \frac{u}{2} \sqrt{2} = \frac{u \sqrt{2}}{2} \]
Calculating \(KE_1\) after the collision:
\[ KE_1' = \frac{1}{2} (2m) \left(\frac{u \sqrt{2}}{2}\right)^2 = \frac{1}{2} (2m) \left(\frac{u^2 \cdot 2}{4}\right) = \frac{m u^2}{2} \]

- Ball 2 (mass = m, final velocity = \(v_2\)):
To find \(v_2\), we use conservation of momentum:
\[ 2m u + m (-2u) = 2m \left(\frac{u}{2} \hat{i} - \hat{j}\right) + m v_2 \]
Calculating the momentum:
\[ 2mu - 2mu = mu \hat{i} - 2m \hat{j} + mv_2 \]
This simplifies to:
\[ 0 = mu \hat{i} - 2m \hat{j} + mv_2 \]
Thus, solving for \(v_2\):
\[ mv_2 = -mu \hat{i} + 2m \hat{j} \implies v_2 = -u \hat{i} + 2u \hat{j} \]
Calculating \(KE_2\) after the collision:
\[ KE_2' = \frac{1}{2} m \left((-u)^2 + (2u)^2\right) = \frac{1}{2} m (u^2 + 4u^2) = \frac{5}{2} mu^2 \]

Thus, the total final kinetic energy is:
\[ KE_{\text{final}} = KE_1' + KE_2' = \frac{mu^2}{2} + \frac{5mu^2}{2} = 3mu^2 \]

### Step 3: Calculate Change in Kinetic Energy
Now, we can find the change in kinetic energy:
\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} = 3mu^2 - 3mu^2 = 0 \]

### Conclusion
The change in kinetic energy of the system is:
\[ \Delta KE = 0 \]