• # question_answer                 Two equal masses ${{m}_{1}}$ and ${{m}_{2}}$ moving along the same straight line with velocities + 3 m/s and ?5 m/s respectively collide elastically. Their velocities after the collision will be respectively: A)                 + 4 m/s for both               B)                 - 3m/s and + 5 m/s          C)                 - 4 m/s and + 4 m/s         D)                 - 5 m/s and + 3 m/s

Solution :

Key Idea: In an elastic collision, linear momentum remains conserved.                 Given: ${{u}_{1}}=3\,m/s,\,{{u}_{2}}=-5m/s,\,{{m}_{1}}={{m}_{2}}=m$                 According to principle of conservation of linear momentum                 ${{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}$                 $m\times 3-m\times 5=m{{v}_{1}}+m{{v}_{2}}$                 or            ${{v}_{1}}+{{v}_{2}}=-2$                                             ...(i)                 In an elastic collision,                 $e=\frac{{{v}_{2}}-{{v}_{1}}}{{{u}_{1}}-{{u}_{2}}}$ $\Rightarrow {{v}_{2}}-{{v}_{1}}=e\,({{u}_{1}}-{{u}_{2}})$ $\Rightarrow {{v}_{2}}-{{v}_{1}}=(1)\,(3+5)$$(\because \,e=1)$ $\Rightarrow {{v}_{1}}-{{v}_{2}}=-8$                                    ...(ii)                 Adding Eqs. (i) and (ii), we obtain                 $2{{v}_{1}}=-\,10\,$                 $\Rightarrow {{v}_{1}}=-5\,m/s$                 From Eq. (i),                 ${{v}_{2}}=-2-{{v}_{1}}=-2+5=3\,m/s$                 Thus, ${{v}_{1}}=-5m/s\,,\,{{v}_{2}}=+3\,m/s$                 Alternative: If two bodies collide elastically, then their velocities are interchanged. Since, it is an elastic collision hence, velocities after collision will be -5 m/s and 3 m/s.

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