First slide

Newton's laws of motion

Question

A bullet of mass m moving with velocity v₀ hits a wooden plank A of mass M placed on a smooth horizontal surface. The length of the plank is $ℓ$ . The bullet experiences a
constant resistive force F inside the block. The minimum value of v₀ such that it is able to come out of the plank is

Moderate

A

$\sqrt{\frac{F ℓ / m}{M^{2}}}$
B

$\sqrt{\frac{2 F ℓ (M + m)}{M m}}$
C

$\sqrt{\frac{2 F ℓ m}{M^{2}}}$
D

$\sqrt{\frac{F ℓ (M + m)}{M m}}$

Solution

From Newton's third law a force F acts on the block in forward direction.
$Acceleration of block a_{1} = \frac{F}{M}$
$Retardation of bullet a_{2} = \frac{F}{m}$
Relative retardation of bullet
$a_{r} = a_{1} + a_{2} = \frac{F (M + m)}{M m}$
$Applying v^{2} = u^{2} - 2 a_{r} ℓ$
$0 = v_{0}^{2} - \frac{2 F (M + m)}{M m} \cdot ℓ$
Therefore, minimum value of v₀ is
$or v_{0} = \sqrt{\frac{2 F ℓ (M + m)}{M m}}$

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