Sure, let's solve this problem step-by-step!
### Step 1: Convert the given masses into kilograms
First, we need to convert the masses from grams to kilograms because the standard unit of mass in the International System of Units (SI) is the kilogram.
- Mass of the bullet [tex]\( m_{\text{bullet}} = 10 \, \text{g} \)[/tex]
[tex]\[
10 \, \text{g} = \frac{10}{1000} \, \text{kg} = 0.01 \, \text{kg}
\][/tex]
- Mass of the plank [tex]\( m_{\text{plank}} = 90 \, \text{g} \)[/tex]
[tex]\[
90 \, \text{g} = \frac{90}{1000} \, \text{kg} = 0.09 \, \text{kg}
\][/tex]
### Step 2: Initial velocities
- Initial velocity of the bullet [tex]\( v_{\text{bullet}} = 1.5 \, \text{m/s} \)[/tex]
- Initial velocity of the plank [tex]\( v_{\text{plank}} = 0 \, \text{m/s} \)[/tex]
### Step 3: Apply the principle of conservation of momentum
When the bullet gets embedded in the plank, the total momentum of the system (bullet + plank) before the collision must equal the total momentum after the collision. The general formula for momentum is [tex]\( p = m \cdot v \)[/tex].
- Initial momentum of the bullet:
[tex]\[
p_{\text{initial, bullet}} = m_{\text{bullet}} \cdot v_{\text{bullet}} = 0.01 \, \text{kg} \cdot 1.5 \, \text{m/s} = 0.015 \, \text{kg} \cdot \text{m/s}
\][/tex]
- Initial momentum of the plank:
[tex]\[
p_{\text{initial, plank}} = m_{\text{plank}} \cdot v_{\text{plank}} = 0.09 \, \text{kg} \cdot 0 \, \text{m/s} = 0 \, \text{kg} \cdot \text{m/s}
\][/tex]
- Total initial momentum of the system:
[tex]\[
p_{\text{initial, total}} = p_{\text{initial, bullet}} + p_{\text{initial, plank}} = 0.015 \, \text{kg} \cdot \text{m/s} + 0 \, \text{kg} \cdot \text{m/s} = 0.015 \, \text{kg} \cdot \text{m/s}
\][/tex]
### Step 4: Calculate the total mass after collision
Both the bullet and the plank move together after the collision, so we need to find the total mass of the combined system.
- Total mass of the system after collision:
[tex]\[
m_{\text{total}} = m_{\text{bullet}} + m_{\text{plank}} = 0.01 \, \text{kg} + 0.09 \, \text{kg} = 0.1 \, \text{kg}
\][/tex]
### Step 5: Determine the final velocity of the system
Using the conservation of momentum, we have:
[tex]\[
p_{\text{initial, total}} = p_{\text{final, total}}
\][/tex]
[tex]\[
0.015 \, \text{kg} \cdot \text{m/s} = m_{\text{total}} \cdot v_{\text{final}}
\][/tex]
Thus we can solve for the final velocity [tex]\( v_{\text{final}} \)[/tex]:
[tex]\[
v_{\text{final}} = \frac{0.015 \, \text{kg} \cdot \text{m/s}}{0.1 \, \text{kg}} = 0.15 \, \text{m/s}
\][/tex]
### Step 6: Conclusion
The final velocity of the bullet embedded in the plank moving together is:
[tex]\[
v_{\text{final}} = 0.15 \, \text{m/s}
\][/tex]
