A particle of mass m moving with speed u collides perfectly inelastically with another particle of mass 3 m at rest. Loss of KE of system in the collision is

Options

A. `3/4mu^(2)`
B. `3/8mu^(2)`
C. `1/4mu^(2)`
D. `1/8mu^(2)`

Answer

To solve the problem of finding the loss of kinetic energy during a perfectly inelastic collision between two particles, we can follow these steps:

### Step 1: Identify the initial conditions
- We have two particles:
- Particle 1 (mass = m) moving with speed u.
- Particle 2 (mass = 3m) at rest (speed = 0).

### Step 2: Calculate the initial kinetic energy (KE_initial)
- The initial kinetic energy of the system is given by:
\[ KE_{\text{initial}} = \frac{1}{2} m u^2 + \frac{1}{2} (3m) (0)^2 = \frac{1}{2} m u^2 \]

### Step 3: Use conservation of momentum to find the final velocity after collision
- According to the law of conservation of momentum:
\[ \text{Total momentum before collision} = \text{Total momentum after collision} \]
- Before the collision:
\[ \text{Total momentum} = m \cdot u + 3m \cdot 0 = mu \]
- After the collision, the two particles stick together, and their combined mass is \( m + 3m = 4m \). Let the final velocity be \( v \):
\[ \text{Total momentum after collision} = (4m) v \]
- Setting the two equal gives:
\[ mu = 4m v \]
- Solving for \( v \):
\[ v = \frac{u}{4} \]

### Step 4: Calculate the final kinetic energy (KE_final)
- The final kinetic energy of the system after the collision is:
\[ KE_{\text{final}} = \frac{1}{2} (4m) v^2 = \frac{1}{2} (4m) \left(\frac{u}{4}\right)^2 \]
- Simplifying this:
\[ KE_{\text{final}} = \frac{1}{2} (4m) \left(\frac{u^2}{16}\right) = \frac{4m u^2}{32} = \frac{m u^2}{8} \]

### Step 5: Calculate the loss of kinetic energy
- The loss of kinetic energy (ΔKE) is given by:
\[ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} \]
- Substituting the values we found:
\[ \Delta KE = \frac{1}{2} m u^2 - \frac{m u^2}{8} \]
- To combine these, we can express \(\frac{1}{2} m u^2\) as \(\frac{4}{8} m u^2\):
\[ \Delta KE = \frac{4}{8} m u^2 - \frac{1}{8} m u^2 = \frac{3}{8} m u^2 \]

### Final Answer
The loss of kinetic energy of the system in the collision is:
\[ \Delta KE = \frac{3}{8} m u^2 \]

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