A body of mass `m_1` moving with an unknown velocity of `v_1 hati` undergoes a collinear collision with a body of mass `m_2` moving with a velocity `v_2 hati`. After collision, `m_1 and m_2` move with velocities of `v_3hati and v_4hati` respectively. If `m_2 = 0.5 m_1 and v_3 = 0.5v_1` then `v_1` is:

Options

A. `v_4 - (v_2)/2`
B. `v_4 - v_2`
C. `v_4 - (v_2)/(4)`
D. `v_4+ v_2`

Explanation

To solve the problem step by step, we need to apply the principle of conservation of linear momentum. Let's break it down:

### Step 1: Write down the conservation of momentum equation.
The total momentum before the collision must equal the total momentum after the collision.

The initial momentum \( P_i \) can be expressed as:
\[ P_i = m_1 v_1 + m_2 v_2 \]

The final momentum \( P_f \) can be expressed as:
\[ P_f = m_1 v_3 + m_2 v_4 \]

Setting these equal gives us:
\[ m_1 v_1 + m_2 v_2 = m_1 v_3 + m_2 v_4 \]

### Step 2: Substitute the known values.
We know from the problem statement that \( m_2 = 0.5 m_1 \) and \( v_3 = 0.5 v_1 \). Substituting these into the equation gives:
\[ m_1 v_1 + 0.5 m_1 v_2 = m_1 (0.5 v_1) + 0.5 m_1 v_4 \]

### Step 3: Simplify the equation.
We can factor out \( m_1 \) from both sides (assuming \( m_1 \neq 0 \)):
\[ v_1 + 0.5 v_2 = 0.5 v_1 + 0.5 v_4 \]

### Step 4: Rearrange the equation.
To isolate \( v_1 \), we can rearrange the equation:
\[ v_1 - 0.5 v_1 = 0.5 v_4 - 0.5 v_2 \]
\[ 0.5 v_1 = 0.5 v_4 - 0.5 v_2 \]

### Step 5: Eliminate the common factor.
Dividing through by \( 0.5 \) gives:
\[ v_1 = v_4 - v_2 \]

### Step 6: Express \( v_4 \) in terms of \( v_1 \).
Since we want \( v_1 \), we need to express \( v_4 \) in terms of \( v_1 \). We can rearrange the equation:
\[ v_4 = v_1 + v_2 \]

### Step 7: Substitute back to find \( v_1 \).
Now we need to find \( v_2 \). From the conservation of momentum, we can express \( v_2 \) in terms of \( v_1 \) and \( v_4 \):
\[ v_2 = v_4 - v_1 \]

### Step 8: Solve for \( v_1 \).
Substituting \( v_4 = 0.5 v_1 + v_2 \) into the equation gives:
\[ v_1 = (0.5 v_1 + v_2) - v_2 \]
This simplifies to:
\[ v_1 = 0.5 v_1 \]

### Step 9: Find \( v_1 \).
This leads us to conclude that:
\[ v_1 = 2v_2 \]

### Final Answer:
Thus, the value of \( v_1 \) can be expressed in terms of \( v_2 \) once we know the value of \( v_2 \).