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`

Explanation

To solve the problem, we need to analyze the elastic collision between two bodies. Let's denote the following:

- Mass of the first body: \( m_1 \)
- Mass of the second body: \( m_2 \)
- Initial velocity of the first body: \( u_1 \)
- Initial velocity of the second body (at rest): \( u_2 = 0 \)
- Final velocity of the first body after collision: \( v_1 = \frac{2}{3} u_1 \)

### Step 1: Write the equation for the final velocity of \( m_1 \) after the elastic collision.

For an elastic collision, the final velocity \( v_1 \) of the first body can be given by the equation:

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

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

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

### Step 2: Substitute the known values into the equation.

We know that \( v_1 = \frac{2}{3} u_1 \). Substituting this into the equation gives:

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

### Step 3: Cancel \( u_1 \) from both sides.

Assuming \( u_1 \neq 0 \), we can cancel \( u_1 \) from both sides:

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

### Step 4: Cross-multiply to eliminate the fraction.

Cross-multiplying gives:

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

### Step 5: Expand and rearrange the equation.

Expanding both sides results in:

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

Now, rearranging the terms gives:

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

This simplifies to:

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

### Step 6: Isolate \( m_1 \) in terms of \( m_2 \).

Rearranging gives:

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

Thus:

\[ m_1 = 5m_2 \]

### Step 7: Find the ratio of the masses.

The ratio of the masses \( \frac{m_1}{m_2} \) is:

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

### Final Answer:

The ratio of their masses is \( 5:1 \).

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