Two particles of masses M and 2 M are moving with speeds of 10 m s - 1 and 5 m s - 1 , as shown in the…

Two particles of masses $M$ and $2\mathrm{M\; are }$moving with speeds of $10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ and $5\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$, as shown in the figure. They collide at the origin and after that they move along the indicated directions with speeds $v_1$ and $v_2$, respectively. The values of $v_1$ and  $v_2$ are, nearly

1. $6.5\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ and $3.2\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$
2. $3.2\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ and $12.6\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$
3. $13.02\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ and $19.7 \mathrm{m} {\mathrm{s}}^{-1}$
4. $3.2\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ and $6.3\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$

Solution

From conservation of linear momentum,
Along $\mathrm{x}$-axis:
$10\mathrm{M}\frac{\sqrt{3}}{2}+10\mathrm{M}\frac{1}{\sqrt{2}}=\mathrm{M}\frac{\sqrt{3}}{2}{\mathrm{v}}_1+2\mathrm{M}{\mathrm{v}}_2\frac{1}{\sqrt{2}}$
$\frac{\sqrt{3}}{2}{\mathrm{v}}_1+\sqrt{2}{\mathrm{v}}_2=5(\sqrt{3}+\sqrt{2})$ … (1)
Along $\mathrm{y}$-axis:
$-10\mathrm{M}\frac{1}{2}+10\mathrm{M}\frac{1}{\sqrt{2}}=\mathrm{M}{\mathrm{v}}_1\frac{1}{2}-2\mathrm{M}{\mathrm{v}}_2\frac{1}{\sqrt{2}}$
$\frac{{\mathrm{v}}_1}{2}-\sqrt{2}{\mathrm{v}}_2=5(\sqrt{2}-1)$ … (2)
From equations (1) and (2)
${\mathrm{v}}_1\left(\frac{\sqrt{3}}{2}+\frac{1}{2}\right)=5\left(\sqrt{3}+\sqrt{2}\right)+5(\sqrt{2}-1)$
${\mathrm{v}}_1=13.02\mathrm{m}/\mathrm{s}$
${\mathrm{v}}_2=\frac{5\sqrt{3}+5\sqrt{2}-13.02\times \frac{\sqrt{3}}{2}}{\sqrt{2}}$
$=19.72\mathrm{m} {\mathrm{s}}^{-1}$

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