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Difficulty: Very hardNewton's Laws of Motion and Linear Momentum

A wooden block of mass 4.0 kg4.0\text{ kg} is suspended vertically at rest. A bullet of mass 0.05 kg0.05\text{ kg} travelling horizontally at 400 m s−1400\text{ m s}^{-1} strikes the block, passes completely through it, and emerges on the opposite side with a reduced speed of 100 m s−1100\text{ m s}^{-1}. If a constant retarding force brings the moving block to rest in 0.25 s0.25\text{ s} after the bullet emerges, calculate the magnitude of this retarding force in newtons.

Answer: 60 N

Answer

The magnitude of the retarding force acting on the block is 60 N60\text{ N}.
During the impact, the bullet loses momentum equal to Δp=0.05 kg×(400 m s−1−100 m s−1)=15 N s\Delta p = 0.05\text{ kg} \times (400\text{ m s}^{-1} - 100\text{ m s}^{-1}) = 15\text{ N s}. By the conservation of linear momentum, this exact amount of momentum is gained by the block. Applying Newton's second law (F=ΔpΔtF = \frac{\Delta p}{\Delta t}), the magnitude of the constant retarding force needed to reduce the block's momentum to zero in 0.25 s0.25\text{ s} is F=15 N s0.25 s=60 NF = \frac{15\text{ N s}}{0.25\text{ s}} = 60\text{ N}.

Step-by-Step Solution

1
Calculate the momentum lost by the bullet during penetration.
Δpbullet=0.05 kg×(400 m s−1−100 m s−1)=15 N s\Delta p_{\text{bullet}} = 0.05\text{ kg} \times (400\text{ m s}^{-1} - 100\text{ m s}^{-1}) = 15\text{ N s}
The momentum lost by the bullet equals its mass multiplied by the change in its horizontal velocity vector.
2
Determine the initial momentum imparted to the wooden block using the law of conservation of linear momentum.
pblock=Δpbullet=15 N sp_{\text{block}} = \Delta p_{\text{bullet}} = 15\text{ N s}
Since no external horizontal force acts during the collision impact, the momentum lost by the bullet is fully transferred to the block.
3
Calculate the retarding force required to bring the block to rest using the impulse-momentum theorem.
F=ΔpblockΔt=15 N s0.25 s=60 NF = \frac{\Delta p_{\text{block}}}{\Delta t} = \frac{15\text{ N s}}{0.25\text{ s}} = 60\text{ N}
According to Newton's second law of motion, the net force acting on a body equals the rate of change of momentum.

Key Concept

Conservation of Linear Momentum and Newton's Second Law
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Topics

Newton's Laws of Motion and Linear Momentum

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