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A block of mass \( m=1 \mathrm{~kg} \), moving on a horizontal surface with speed \( v_{i}=2 \mathrm{~ms}^{-1} \) enters a rough patch ranging from \( x=0.10 \mathrm{~m} \) to \( x=2.01 \mathrm{~m} \). The retarding force \( F_{r} \) on the block in this range is inversely proportional to \( x \) over this range,
\( F_{r}=\frac{-k}{x} \) for \( 0.1x2.01 \mathrm{~m} \)
\( =0 \) for \( x0.1 \mathrm{~m} \) and \( x2.01 \mathrm{~m} \), where \( k=0.5 \mathrm{~J} \).
What is the speed \( v_{f} \) of the block as it crosses this patch?
(a) \( 2 \mathrm{~ms}^{-1} \)
(b) \( 40 \mathrm{~ms}^{-1} \)
(c) \( 1 \mathrm{~ms}^{-1} \)
(d) \( 4 \mathrm{~ms}^{-1} \)
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