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4-2.Friction
hard
A sphere of mass $m$ is set in motion with initial velocity $v_o$ on a surface on which $kx^n$ is the frictional force with $k$ and $n$ as the constants and $x$ as the distance from the point of start. Find the distance in which sphere will stop
A
${\left[ {\frac{{m{v^2}_0(n\, + \,1)}}{{2k}}} \right]^{1/(n + 1)}}$
B
${\left[ {\frac{{m{v^2}_0}}{{2k}}} \right]^{1/(n - 1)}}$
C
${\left[ {\frac{{2m{v^2}_0}}{{k}}} \right]^{1/(n - 1)}}$
D
${\left[ {\frac{{m{v^2}_0}}{{2k(n - 1)}}} \right]^{1/(n - 1)}}$
Solution
$F=K x^{n}$
$-m \frac{v d v}{d x}=K x^{n}$
$-m \int_{v_{0}}^{0} v d v=K \int_{0}^{x} x^{n} d x$
$\Rightarrow \frac{m v_{0}^{2}}{2}=\frac{K x^{n+1}}{n+1}$
$\Rightarrow x=\left[\frac{m v_{0}^{2}(n+1)}{2 K}\right]^{\frac{1}{n+1}}$
Standard 11
Physics
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