Two identical particles collide in air inelastically. One moves horizontally and the other moves vertically with equal speed just before collision. The fractional loss in kinetic energy of the system of particles is equal to

Options

A. `1/sqrt(2)`
B. `1/2`
C. `sqrt(3)/2`
D. none of these

Explanation

To solve the problem of two identical particles colliding inelastically, we will follow these steps:

### Step 1: Define the initial conditions
Let each particle have a mass \( m \) and both are moving with the same speed \( v \). One particle is moving horizontally, and the other is moving vertically.

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (\( KE_{initial} \)) of the system can be calculated as:
\[ KE_{initial} = \frac{1}{2} mv^2 + \frac{1}{2} mv^2 = mv^2 \]

### Step 3: Apply conservation of momentum
Since the collision is inelastic, the two particles stick together after the collision. We will apply the conservation of momentum in both the x and y directions.

- In the x-direction:
\[ m v + 0 = (2m) u_x \quad \text{(where \( u_x \) is the x-component of the final velocity)} \]
This simplifies to:
\[ mv = 2m u_x \implies u_x = \frac{v}{2} \]

- In the y-direction:
\[ 0 + mv = (2m) u_y \quad \text{(where \( u_y \) is the y-component of the final velocity)} \]
This simplifies to:
\[ mv = 2m u_y \implies u_y = \frac{v}{2} \]

### Step 4: Calculate the final velocity
The final velocity \( u \) of the combined mass can be found using the Pythagorean theorem:
\[ u = \sqrt{u_x^2 + u_y^2} = \sqrt{\left(\frac{v}{2}\right)^2 + \left(\frac{v}{2}\right)^2} = \sqrt{\frac{v^2}{4} + \frac{v^2}{4}} = \sqrt{\frac{v^2}{2}} = \frac{v}{\sqrt{2}} \]

### Step 5: Calculate the final kinetic energy
The final kinetic energy (\( KE_{final} \)) of the system after the collision is:
\[ KE_{final} = \frac{1}{2} (2m) u^2 = m \left(\frac{v}{\sqrt{2}}\right)^2 = m \cdot \frac{v^2}{2} = \frac{mv^2}{2} \]

### Step 6: Calculate the loss in kinetic energy
The loss in kinetic energy (\( \Delta KE \)) is given by:
\[ \Delta KE = KE_{initial} - KE_{final} = mv^2 - \frac{mv^2}{2} = \frac{mv^2}{2} \]

### Step 7: Calculate the fractional loss in kinetic energy
The fractional loss in kinetic energy is given by:
\[ \text{Fractional loss} = \frac{\Delta KE}{KE_{initial}} = \frac{\frac{mv^2}{2}}{mv^2} = \frac{1}{2} \]

### Conclusion
Thus, the fractional loss in kinetic energy of the system of particles is \( \frac{1}{2} \).