To solve the problem step by step, we will use the concepts of momentum and the center of mass.
### Step 1: Understand the given information
We have:
- Total momentum of the system, \( P = 18 \, \text{kg m/s} \, \hat{i} \)
- Velocity of the center of mass, \( V_{cm} = 3 \, \text{m/s} \, \hat{i} \)
- Mass of the first object, \( m_1 = 4 \, \text{kg} \)
- Velocity of the first object, \( v_1 = 1.5 \, \text{m/s} \, \hat{i} \)
We need to find the mass of the second object, \( m_2 \).
### Step 2: Use the formula for total momentum
The total momentum of the system is given by the sum of the momenta of both objects:
\[
P = m_1 v_1 + m_2 v_2
\]
Substituting the known values:
\[
18 = 4 \times 1.5 + m_2 v_2
\]
Calculating \( 4 \times 1.5 \):
\[
4 \times 1.5 = 6
\]
So, we have:
\[
18 = 6 + m_2 v_2
\]
This simplifies to:
\[
m_2 v_2 = 18 - 6 = 12
\]
Thus, we have:
\[
m_2 v_2 = 12 \quad \text{(1)}
\]
### Step 3: Use the formula for the velocity of the center of mass
The velocity of the center of mass is given by:
\[
V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}
\]
Substituting the known values:
\[
3 = \frac{4 \times 1.5 + m_2 v_2}{4 + m_2}
\]
We already calculated \( 4 \times 1.5 = 6 \), so:
\[
3 = \frac{6 + m_2 v_2}{4 + m_2}
\]
Cross-multiplying gives:
\[
3(4 + m_2) = 6 + m_2 v_2
\]
Expanding this:
\[
12 + 3m_2 = 6 + m_2 v_2
\]
Rearranging gives:
\[
3m_2 - m_2 v_2 = 6 - 12
\]
This simplifies to:
\[
3m_2 - m_2 v_2 = -6
\]
Factoring out \( m_2 \):
\[
m_2(3 - v_2) = -6 \quad \text{(2)}
\]
### Step 4: Solve equations (1) and (2)
From equation (1), we have:
\[
v_2 = \frac{12}{m_2}
\]
Substituting this into equation (2):
\[
m_2 \left( 3 - \frac{12}{m_2} \right) = -6
\]
This simplifies to:
\[
m_2 \cdot 3 - 12 = -6
\]
So:
\[
3m_2 = 6
\]
Thus:
\[
m_2 = 2 \, \text{kg}
\]
### Final Answer
The mass of the other object is \( \boxed{2} \, \text{kg} \).
