A body of mass `m_(1) ` moving with uniform velocity of 40 m/s collides with another mass `m_(2)` at rest and then the two together begin to moe wit h uniform velocity of 30 m/s. the ratio of their masses `(m_(1))/(m_(2))` is

Options

A. `1:3`
B. `3:1`
C. `1:1.33`
D. `1:0.75`

Explanation

To solve the problem, we will use the principle of conservation of momentum. Let's break down the steps:

### Step 1: Understand the Given Information
- Mass \( m_1 \) is moving with a velocity \( u_1 = 40 \, \text{m/s} \).
- Mass \( m_2 \) is at rest, so its initial velocity \( u_2 = 0 \, \text{m/s} \).
- After the collision, both masses move together with a final velocity \( v = 30 \, \text{m/s} \).

### Step 2: Write the Conservation of Momentum Equation
According to the law of conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
This can be expressed as:
\[ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v \]

### Step 3: Substitute the Known Values
Substituting the known values into the equation:
\[ m_1 \cdot 40 + m_2 \cdot 0 = (m_1 + m_2) \cdot 30 \]
This simplifies to:
\[ 40 m_1 = (m_1 + m_2) \cdot 30 \]

### Step 4: Expand the Right Side
Expanding the right side gives:
\[ 40 m_1 = 30 m_1 + 30 m_2 \]

### Step 5: Rearrange the Equation
Rearranging the equation to isolate terms involving \( m_1 \) and \( m_2 \):
\[ 40 m_1 - 30 m_1 = 30 m_2 \]
This simplifies to:
\[ 10 m_1 = 30 m_2 \]

### Step 6: Solve for the Ratio of Masses
Dividing both sides by \( m_2 \) and \( 10 \):
\[ \frac{m_1}{m_2} = \frac{30}{10} = 3 \]
Thus, the ratio of the masses \( \frac{m_1}{m_2} \) is \( 3:1 \).

### Final Answer
The ratio of the masses \( \frac{m_1}{m_2} \) is \( 3:1 \).

---