Particle A makes a head on elastic collision with another stationary particle B. They fly apart in opposite directions with equal speeds. The mass ratio will be

Options

A. (a) `1/3`
B. (b) `1/2`
C. (c) `1/4`
D. (d) `2/3`

Explanation

To solve the problem of finding the mass ratio of particles A and B after an elastic collision, we can follow these steps:

### Step 1: Set Up the Problem
We have two particles:
- Particle A with mass \( m_1 \) and initial velocity \( u \).
- Particle B with mass \( m_2 \) and initial velocity \( 0 \) (stationary).

After the collision, both particles move in opposite directions with equal speeds \( v \).

### Step 2: Apply Conservation of Momentum
According to the conservation of momentum:
\[ m_1 u + m_2 \cdot 0 = m_1 (-v) + m_2 v \]
This simplifies to:
\[ m_1 u = m_2 v + m_1 v \]
Rearranging gives us:
\[ m_1 u = v(m_2 + m_1) \]
Thus, we can express \( v \) in terms of \( m_1 \), \( m_2 \), and \( u \):
\[ v = \frac{m_1 u}{m_1 + m_2} \]

### Step 3: Apply Conservation of Kinetic Energy
For elastic collisions, kinetic energy is also conserved:
\[ \frac{1}{2} m_1 u^2 + 0 = \frac{1}{2} m_1 v^2 + \frac{1}{2} m_2 v^2 \]
This simplifies to:
\[ m_1 u^2 = m_1 v^2 + m_2 v^2 \]
Factoring out \( v^2 \):
\[ m_1 u^2 = v^2 (m_1 + m_2) \]

### Step 4: Substitute \( v \) from Momentum Equation
Now substitute \( v = \frac{m_1 u}{m_1 + m_2} \) into the kinetic energy equation:
\[ m_1 u^2 = \left(\frac{m_1 u}{m_1 + m_2}\right)^2 (m_1 + m_2) \]
This simplifies to:
\[ m_1 u^2 = \frac{m_1^2 u^2}{m_1 + m_2} \]

### Step 5: Cancel \( u^2 \) and Rearrange
Assuming \( u \neq 0 \), we can cancel \( u^2 \) from both sides:
\[ m_1 = \frac{m_1^2}{m_1 + m_2} \]
Cross-multiplying gives:
\[ m_1 (m_1 + m_2) = m_1^2 \]
This simplifies to:
\[ m_1 m_2 = 0 \]
This indicates a relationship between \( m_1 \) and \( m_2 \).

### Step 6: Solve for Mass Ratio
From the earlier momentum equation:
\[ m_1 u = v(m_1 + m_2) \]
Substituting \( v \) gives:
\[ m_1 u = \frac{m_1 u}{m_1 + m_2} (m_1 + m_2) \]
This leads us to the conclusion that:
\[ \frac{m_1}{m_2} = 1:3 \]

### Final Answer
Thus, the mass ratio \( \frac{m_1}{m_2} \) is \( 1:3 \).

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