Center of Mass, Momentum & Collision

JEE Main level, single correct, medium

111 of 130

Question 111 of 130

Two particles of equal mass mm have respective initial velocities ui^u\hat{i} and u(i^+j^2)u\left(\dfrac{\hat{i}+\hat{j}}{2}\right) . They collide completely inelastically. The energy lost in the process is:

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Answer: (B) 18mu2\dfrac{1}{8}m{u}^{2}

Conserving momentum
mui^+m(u2i^+u2j^)=2m(u1i^+u2j^)mu\hat{i}+m\left(\dfrac{u}{2}\hat{i}+\dfrac{u}{2}\hat{j}\right)=2m\left({u}_{1}\hat{i}+{u}_{2}\hat{j}\right)
On solving
u1=3u4{u}_{1}=\dfrac{3u}{4} and u2=u4{u}_{2}=\dfrac{u}{4}
Change in K.E.
[12mu2+12m(u22)2]−[12(2m)(9u216+u216)]\left[\dfrac{1}{2}m{u}^{2}+\dfrac{1}{2}m{\left(\dfrac{u}{2}\sqrt{2}\right)}^{2}\right]-\left[\dfrac{1}{2}\left(2m\right)\left(\dfrac{9{u}^{2}}{16}+\dfrac{{u}^{2}}{16}\right)\right]
=3mu24−5mu28=mu28=\dfrac{3m{u}^{2}}{4}-\dfrac{5m{u}^{2}}{8}=\dfrac{m{u}^{2}}{8}

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