Two bodies of masses $m_{1}$ and $m_{2}$ moving on the same direction with velocities $u_{1}$ and $u_{2}$ collide. The velocities after collision are $v_{1}$ and $v_{2}$ . If each sphere loses the same amount of kinetic energy, then

u_{1} + u_{2} + v_{1} - v_{2} = 0

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u_{1} - u_{2} + v_{1} + v_{2} = 0

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u_{1} + u_{2} + v_{1} + v_{2} = 0

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u_{1} + u_{2} - v_{1} - v_{2} = 0

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Solution

The correct option is
D
$u_{1} + u_{2} + v_{1} + v_{2} = 0$

Using momentum conservation:
$m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ .........(1)
Using conservation of kinetic energy we get
$\frac{1}{2} m_{2} {u_{2}}^{2} - \frac{1}{2} m_{2} {v_{2}}^{2} =$
$\frac{1}{2} m_{1} {u_{1}}^{2} - \frac{1}{2} m_{1} {v_{1}}^{2}$
$\frac{1}{2} m_{2} ({u_{2}}^{2} - {v_{2}}^{2}) = \frac{1}{2} m_{1} ({u_{1}}^{2} - {v_{1}}^{2})$
From (1):
$\frac{1}{2} (u_{2} + v_{2}) \times m_{1} (v_{1} - u_{1}) = \frac{1}{2} m_{1} ({u_{1}}^{2} - {v_{1}}^{2})$
$\Rightarrow u_{2} + v_{2} = - u_{1} - v_{1}$
$\Rightarrow u_{1} + v_{1} + u_{2} + v_{2} = 0$

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The correct option is D u1+u2+v1+v2=0Using momentum conservation:m1u1+m2u2=m1v1+m2v2.........(1)Using conservation of kinetic energy we get12m2u22−12m2v22=12m1u12−12m1v1212m2(u22−v22)=12m1(u12−v12)From (1):12(u2+v2)×m1(v1−u1)=12m1(u12−v12)⇒u2+v2=−u1−v1⇒u1+v1+u2+v2=0

Two bodies of masses $m_{1}$ and $m_{2}$ moving on the same direction with velocities $u_{1}$ and $u_{2}$ collide. The velocities after collision are $v_{1}$ and $v_{2}$ . If each sphere loses the same amount of kinetic energy, then