A) \[\frac{m{{v}_{0}}^{2}}{{{x}_{0}}^{2}}\]
B) \[\frac{m{{v}_{0}}^{2}}{2{{x}_{0}}^{2}}\]
C) \[\frac{3}{2}\frac{m{{v}_{0}}^{2}}{{{x}_{0}}^{2}}\]
D) \[\frac{2}{3}\frac{m{{v}_{0}}^{2}}{{{x}_{0}}^{2}}\]
Correct Answer: D
Solution :
Key Idea: In an elastic collision, the conservation of linear momentum and conservation of energy hold. Using conservation of linear momentum, we have \[m{{v}_{0}}=mv+2mv\] \[\Rightarrow \] \[v=\frac{{{v}_{0}}}{3}\] Using conservation of energy, we have \[\frac{1}{2}mv_{0}^{2}=\frac{1}{2}kx_{0}^{2}+\frac{1}{2}(3m){{v}^{2}}\] where \[{{x}_{0}}\] is compression in the string. \[\therefore \] \[mv_{0}^{2}+kx_{0}^{2}+(3m)\frac{v_{0}^{2}}{9}\] \[\Rightarrow \] \[kx_{0}^{2}=mv_{0}^{2}-\frac{mv_{0}^{2}}{3}\] \[\Rightarrow \] \[kx_{0}^{2}=\frac{2mv_{0}^{2}}{3}\] \[\therefore \] \[k=\frac{2mv_{0}^{2}}{3x_{0}^{2}}\] Note: In an inelastic collision between two bodies, only conservation of linear momentum hold.
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