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A bullet of mass $m$ strikes a block of mass $M$ connected to a light spring of stiffness $k,$ with a speed $v_0.$ If the bullet gets embedded in the block then, the maximum compression in the spring is
${\left( {\frac{{{m^2}v_0^2}}{{(M + m)k}}} \right)^{1/2}}$
${\left( {\frac{{Mmv_0^2}}{{2(M + m)k}}} \right)^{1/2}}$
${\left( {\frac{{mv_0^2}}{{2(M + m)k}}} \right)^{1/2}}$
${\left( {\frac{{Mv^2}}{{(M + m)k}}} \right)^{1/2}}$
Solution
By $COLM$
$\mathrm{mv}_{0}+0=(\mathrm{m}+\mathrm{M}) \mathrm{V}$
$\mathrm{V}=\frac{\mathrm{m}}{\mathrm{m}+\mathrm{M}} \mathrm{V}_{0}$ $…(1)$
By $CONE$
$\frac{1}{2}(\mathrm{m}+\mathrm{M}) \mathrm{V}^{2}=\frac{1}{2} \mathrm{kx}^{2}$ $…(2)$
by solving $(1)$ and $(2)$
$\mathrm{x}=\sqrt{\frac{\mathrm{m}^{2} \mathrm{v}_{0}^{2}}{(\mathrm{m}+\mathrm{M}) \mathrm{k}}}$
