- #1

Saw

Gold Member

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My questions:In any collision:

[tex]

\frac{{m_1 }}{{m_2 }} = \frac{{a_2 }}{{a_1 }} = \frac{{\frac{{\Delta v_2 }}{t}}}{{\frac{{\Delta v_1 }}{t}}} = \frac{{\Delta v_2 }}{{\Delta v_1 }}

[/tex]

If the system formed by the two bodies is closed (no external force) and the collision is perfectly elastic (no internal dissipation of energy), then the relative speed of the two bodies does not change. For example, in the frame of m1, m2 was approaching at a certain v and after the collision it recedes in the opposite direction. This means that:

[tex]

\Delta v_1 + \Delta v_2 = 2v_{rel}

[/tex]

After some algebra:

[tex]

\begin{array}{l}

\frac{{m_1 }}{{m_2 }} + 1 = \frac{{\Delta v_2 }}{{\Delta v_1 }} + 1 = \frac{{m_1 + m_2 }}{{m_2 }} = \frac{{\Delta v_2 + \Delta v_1 }}{{\Delta v_1 }} = \frac{{2v_{rel} }}{{\Delta v_1 }} \to \Delta v_1 = 2v_{rel} \frac{{m_2 }}{{m_1 + m_2 }} \\

\frac{{m_2 }}{{m_1 }} + 1 = \frac{{\Delta v_1 }}{{\Delta v_2 }} + 1 = \frac{{m_2 + m_1 }}{{m_1 }} = \frac{{\Delta v_1 + \Delta v_2 }}{{\Delta v_2 }} = \frac{{2v_{rel} }}{{\Delta v_2 }} \to \Delta v_2 = 2v_{rel} \frac{{m_1 }}{{m_1 + m_2 }} \\

\end{array}

[/tex]

(a) I haven't seen this approach in the texts I use. Is this correct? Am I missing anything?

(b) If correct, is it also valid for acceleration, not only for velocity increase? It seems to me so, since the interaction (the collision) will take the time it takes, whatever it is, but in any case the same time for both bodies and just after it, after the bodies start separating, there is no interaction any more, so no more acceleration...