Sure, let's tackle this problem step by step using the principles of physics, specifically the conservation of momentum.
Step 1: Understand the problem setup
- We have a bullet of mass [tex]\( m \)[/tex] traveling with a velocity [tex]\( v \)[/tex].
- The bullet hits a wooden block of mass [tex]\( M \)[/tex], which is initially at rest.
- After the collision, the bullet gets embedded in the wooden block, and they move together as a single composite system.
Step 2: Recognize the conservation of momentum principle
- Momentum is conserved in an isolated system (no external forces acting on the system).
- Before the collision, the momentum is purely due to the bullet since the block is at rest.
- Initial momentum of the system: [tex]\( m \cdot v \)[/tex]
- After the collision, both the bullet and the wooden block move together with a combined mass [tex]\( (m + M) \)[/tex] and a common velocity [tex]\( V_{\text{final}} \)[/tex].
- Final momentum of the system: [tex]\( (m + M) \cdot V_{\text{final}} \)[/tex]
Step 3: Set up the conservation of momentum equation
According to the conservation of momentum:
[tex]\[ \text{Initial momentum} = \text{Final momentum} \][/tex]
[tex]\[ m \cdot v = (m + M) \cdot V_{\text{final}} \][/tex]
Step 4: Solve for the final velocity [tex]\( V_{\text{final}} \)[/tex]
To find [tex]\( V_{\text{final}} \)[/tex], we rearrange the equation:
[tex]\[ V_{\text{final}} = \frac{m \cdot v}{m + M} \][/tex]
Step 5: Interpret the results
- This equation shows that the final velocity of the composite system (the bullet embedded in the block) depends on the masses involved and the initial velocity of the bullet.
- The correct option from the given choices is:
[tex]\[ \boxed{\frac{m \cdot v}{M + m}} \][/tex]
Thus, the velocity of the composite system after the collision is given by option (D) [tex]\(\frac{m v}{ M + m}\)[/tex].
