two perfectly elastic spheres of masses 4kg & 6kg moving in opposite directions with velocities of 4ms^-1 & 6ms^-1 collide with each other. their velocities after head on collision are? (ans: -8ms^-1 , 2ms^-1)

Dear Student,

Please find below the solution to the asked query:

By using conservation of momentum and conservation of energy, the equations for the velocities of the spheres after collision are,

$$\begin{gathered} V_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)U_1+\left(\frac{2m_2}{m_1+m_2}\right)U_2 \\ and \\ {V}_2=\left(\frac{m_2-m_1}{m_1+m_2}\right)U_2+\left(\frac{2m_1}{m_1+m_2}\right)U_1 \end{gathered}$$

Where,
m₁ = mass of the first sphere (= 4 kg)
m₂ = mass of the second sphere (=6 kg)
U₁ = speed of first sphere before collision (=4 m/s)
U₂ = speed of second sphere before collision (= - 6 m/s)
V₁ = speed of first sphere after collision
V₂ = speed of second sphere after collision

Therefore,

$$\begin{gathered} V_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)U_1+\left(\frac{2m_2}{m_1+m_2}\right)U_2 = \left(\frac{4-6}{4+6}\right)4+\left(\frac{2\times 6}{4+6}\right)\left(-6\right) \\ \Rightarrow V_1= \frac{-8}{10}-\frac{72}{10}=\frac{-80}{10} \\ \Rightarrow V_1=-8 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$
and
$$\begin{gathered} V_2=\left(\frac{m_2-m_1}{m_1+m_2}\right)U_2+\left(\frac{2m_1}{m_1+m_2}\right)U_1 = \left(\frac{6-4}{4+6}\right)\left(-6\right)+\left(\frac{2\times 4}{4+6}\right)\left(4\right) \\ \Rightarrow V_2= \frac{-12}{10}+\frac{32}{10}=\frac{20}{10} \\ \Rightarrow V_2=+2 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

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