(adsbygoogle = window.adsbygoogle || []).push({}); Hi all,

1. The problem statement, all variables and given/known data

I'm stupefied. In continuation to my last thread (https://www.physicsforums.com/showthread.php?t=277124), I wanted to see if any perfectly plastic collision would make the total mechanical energy of the system go down to a factor of [tex]\frac{1}{\sqrt 6}[/tex]. I got a strange result and I know it cannot be possible so there's at least one error. If you could find it out... I'd be grateful.

Say you have initially 2 particles in motion. The first one has a mass [tex]m[/tex] and a speed of [tex]v_1[/tex]. Second one has a mass [tex]M[/tex] and a speed of [tex]v_2[/tex].

They collide and after the collision they remain attached. They form a new particle of mass [tex]M+m[/tex] and speed [tex]v_3[/tex]. (I assume speed instead of velocity to simplify but I might be wrong by doing this however).

The linear momentum is conserved. Putting away vectors, I have that [tex]P_i=P_f \Leftrightarrow mv_1+Mv_2=(M+m)v_3 \Leftrightarrow v_3=\frac{mv_1+Mv_2}{M+m}[/tex].

As the mechanical energy is not conserved I still want to find out the energy before and after the collision in order to compare them.

I have that [tex]E_i=\frac{mv_1^2+Mv_2^2}{2}[/tex] while [tex]E_f=\frac{(m+M)v_3^2}{2}=\frac{(mv_1+Mv_2)^2}{2(m+M)}[/tex].

I want to know how much times the initial energy is greater than the final. So [tex]E_i=\alpha E_f \Leftrightarrow \alpha= \frac{mv_1^2+Mv_2^2}{2} \cdot \frac{2(m+M)}{(mv_1+Mv_2)^2}[/tex] by expanding I finally get that [tex]\alpha=\frac{m^2v_1^2+Mmv_2^2+Mmv_1^2+M^2v_2^2}{m^2v_1^2+2mv_1Mv_2+M^2v_2^2}[/tex]. Note that the numerator and the denominator are almost equal, they only differ by the term "[tex]Mm(v_2^2+v_1^2)[/tex]" in the numerator and "[tex]2mv_1Mv_2[/tex]" at the denominator. But if you set [tex]v_1[/tex] and [tex]v_2[/tex] to be [tex]\frac {1m}{s} }[/tex], there are equal and as a consequence the mechanical energy is conserved...which is obviously wrong. Where did I go wrong?

Thank you very much.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Loss of mechanical energy in a plastic collision

**Physics Forums | Science Articles, Homework Help, Discussion**