\(\,\)
\psset{xunit=1.5cm,yunit=1.5cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.3pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-2.73,-2.6)(3.4,1.82)
\psline{->}(-2.73,0)(2.5,0)
\psline{->}(0,-2.2)(0,1.5)
\rput[tl](2.55,0.05){\(x\)}
\rput[tl](-0.05,1.75){\(y\)}
\rput[tl](-2.09,-0.3){\(P_1\)} \rput[tl](-1.11,-0.3){\(P_2\)} \rput[tl](-0.55,-0.6){\(P_3\)} \rput[tl](-0.55,-1.6){\(P_4\)} \rput[tl](0.07,0.3){\(O\)} \psline{->}(-2.1,0.4)(-1.6,0.4) \psline{->}(0.4,-1.9)(0.4,-1.4) \rput[tl](-1.55,0.45){\(u\)} \rput[tl](0.32,-1.2){\(u\)} \begin{scriptsize} \psdots[dotsize=18pt 0,dotstyle=*](-1,0) \psdots[dotsize=18pt 0,dotstyle=*](-2,0) \psdots[dotsize=18pt 0,dotstyle=*](0,-0.7) \psdots[dotsize=18pt 0,dotstyle=*](0,-1.7) \end{scriptsize} \end{pspicture*}
Four particles \(P_1\), \(P_2\), \(P_3\) and
\(P_4\), of masses \(m_1\), \(m_2\), \(m_3\) and \(m_4\), respectively, are arranged
on smooth horizontal axes as shown in the diagram.
Initially, \(P_2\) and \(P_3\) are stationary, and
both \(P_1\) and \(P_4\) are moving towards \(O\) with speed \(u\).
Then \(P_1\) and \(P_2\) collide, at the same moment as
\(P_4\) and \(P_3\) collide. Subsequently, \(P_2\) and
\(P_3\) collide at \(O\), as do \(P_1\) and \(P_4\) some time later.
The coefficient of
restitution between each pair of particles is~\(e\), and \(e>0\).
Show that initially \(P_2\) and \(P_3\) are equidistant from \(O\).