Question : A bullet of mass 'm' and velocity 'a' is fired into a large block of wood of mass 'M'. The final velocity of the system is:
Option 1: $\frac{M}{m + M}a$
Option 2: $\frac{\vec{m}}{ m + M}a$
Option 3: $\frac{m + M}{m}a$
Option 4: $\frac{m + M}{M}a$
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Correct Answer: $\frac{\vec{m}}{ m + M}a$
Solution : The correct option is $\frac{\vec{m}}{ m + M}a$ .
The initial momentum of the bullet is 'ma' (mass 'm' times velocity 'a'). The initial momentum of the block is 0 since it is initially at rest.
After the collision, the bullet becomes embedded in the block, and they move together as one system. Let's denote their final velocity as 'V'. The total mass of the system (bullet + block) is +M'.
According to the law of conservation of linear momentum:
ma = (m + M) * V
V = (ma)/(m + M).
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