Linear Momentum
Momentum \( \vec{p} = m \vec{V} \), in kg.m/s
Rewrite N2L as
\[
\vec{F} = \lim_{\Delta t \rightarrow 0} {{\Delta \vec{p}}\over{\Delta t}}
\quad \mbox{or} \quad
\sum \vec{F}_{\rm ext} = \lim_{\Delta t \rightarrow 0} {{\Delta \vec{p}_{\rm total}}\over{\Delta t}}
\]
Total impulse \( {\cal I} = F \, \Delta t = \Delta p = \Delta (mV) \)
is the area under the curve on the \(F\) vs. \(t\) graph
N2L as the law of conservation of momentum
\[
\sum \vec{F}_{\rm ext}=0 \quad \rightarrow \quad \Delta \vec{p}_{\rm total} = 0 \quad \rightarrow \quad \vec{p}_{\rm total} ={\rm const}
\]
Conservation Laws in solving problems
Elastic collision C.o.M. and C.o.K.E.
\begin{eqnarray*}
\vec{V}_{f,1} & = & {\frac{m_1 - m_2}{m_1 + m_2}} \, \vec{V}_{i,1} \quad + \quad \frac{2m_2}{m_1 + m_2} \, \vec{V}_{i,2} \\
\vec{V}_{f,2} & = & {\frac{2m_1}{m_1 + m_2}} \, \vec{V}_{i,1} \quad + \quad \frac{m_2 - m_1}{m_1 + m_2} \, \vec{V}_{i,2}
\end{eqnarray*}
Inelastic collision $\Delta E\neq0$, but C.o.M. holds!
\[
\vec{V}_f \quad = \quad {{m_1}\over{m_1 + m_2}} \, \vec{V}_{i,1}
\quad + \quad {{m_2}\over{m_1 + m_2}} \, \vec{V}_{i,2}
\]
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