A body of mass `m_(1)` collides elastically with another body of mass `m_(2)` at rest. If the velocity of `m_(1)` after collision is `(2)/(3)` times its initial velocity, the ratio of their masses is `:`

Options

A. `1:5`
B. `5:1`
C. `5:2`
D. `2:5`

Answer

To solve the problem step by step, we will use the principles of conservation of momentum and the properties of elastic collisions.

### Step 1: Understand the scenario
We have two bodies:
- Body 1 with mass \( m_1 \) is moving with initial velocity \( u_1 \).
- Body 2 with mass \( m_2 \) is at rest, so its initial velocity \( u_2 = 0 \).

### Step 2: Write down the equations for elastic collision
In an elastic collision, both momentum and kinetic energy are conserved. However, we will use the formula for final velocities after an elastic collision:

\[ v_1 = \frac{m_1 - m_2}{m_1 + m_2} u_1 + \frac{2m_2}{m_1 + m_2} u_2 \]

Since \( u_2 = 0 \), the equation simplifies to:

\[ v_1 = \frac{m_1 - m_2}{m_1 + m_2} u_1 \]

### Step 3: Use the given information
We know that the velocity of \( m_1 \) after the collision is \( v_1 = \frac{2}{3} u_1 \). Substituting this into our equation gives:

\[ \frac{2}{3} u_1 = \frac{m_1 - m_2}{m_1 + m_2} u_1 \]

### Step 4: Cancel \( u_1 \) from both sides
Assuming \( u_1 \neq 0 \), we can divide both sides by \( u_1 \):

\[ \frac{2}{3} = \frac{m_1 - m_2}{m_1 + m_2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:

\[ 2(m_1 + m_2) = 3(m_1 - m_2) \]

### Step 6: Expand and rearrange the equation
Expanding both sides results in:

\[ 2m_1 + 2m_2 = 3m_1 - 3m_2 \]

Rearranging the terms gives:

\[ 2m_1 + 2m_2 + 3m_2 = 3m_1 \]

This simplifies to:

\[ 2m_1 + 5m_2 = 3m_1 \]

### Step 7: Isolate \( m_1 \)
Rearranging gives:

\[ 5m_2 = 3m_1 - 2m_1 \]

So we have:

\[ 5m_2 = m_1 \]

### Step 8: Find the ratio of the masses
To find the ratio \( \frac{m_1}{m_2} \):

\[ \frac{m_1}{m_2} = 5 \]

Thus, the ratio of their masses is \( 5:1 \).

### Conclusion
The final answer is:

\[ \text{Ratio of } m_1 \text{ to } m_2 = 5:1 \]

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