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Block enters a rough region where the retarding force is F = -kx: speed or energy after crossing it

The same question, asked 2 times: 28 Jun 2022*, 3 Apr 2025. Every wording below, oldest first.

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  1. Question 1

    28 Jun 2022

    93
    A block of mass 2 kg2\text{ }kg moving on a horizontal surface with speed of 4 ms−14{\text{ }ms}^{- 1} enters a rough surface ranging from x=0.5 mx = 0.5\text{ }m to =1.5 m= 1.5\text{ }m. The retarding force in this range of rough surface is related to distance by F=−kxF = - kx where k=12Nm−1k = 12Nm^{- 1}. The speed of the block as it just crosses the rough surface will be:

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  2. Question 2

    3 Apr 2025

    94
    A block of mass 1 kg , moving along xx with speed vi=10 m/sv_{i}= 10\text{ }m/s enters a rough region ranging from x=0.1 mx = 0.1\text{ }m to x=1.9 mx = 1.9\text{ }m. The retarding force acting on the block in this range is Fr=−kxNF_{r}= - kxN, with k=10 N/mk = 10\text{ }N/m. Then the final speed of the block as it crosses rough region is.

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