= Crash pulse
{title2=$a(t)$}
A \b[crash pulse] is an <acceleration> time history associated with a <collision>. It must specify a measurement location, direction and reference frame: a vehicle structure, test platform and supported component can have different histories. NASA's 1981 aircraft-seat study explicitly described crash pulses as deceleration time histories and used them to inform <crashworthiness> testing.<Determination of crash test pulses and their application to aircraft seat analysis by E. Alfaro-Bou and others>[NASA NTRS19810058363, author abstract]{ref}
For one-dimensional motion, the signed area under the <acceleration> curve is the change in <velocity>:
$$
\Delta v=\int_0^T a(t)\,dt.
$$
Consider two original ideal stopping pulses, both lasting $T=0.10\,\mathrm s$ and reducing velocity from $8\,\mathrm{m\,s^{-1}}$ to zero. A constant pulse needs $a=-80\,\mathrm{m\,s^{-2}}$. A symmetric triangular pulse that begins and ends at zero needs a peak magnitude of $160\,\mathrm{m\,s^{-2}}$, because its area is half its peak times its duration:
$$
\Delta v_{\rm constant}=(-80)(0.10)=-8\,\mathrm{m\,s^{-1}},
\qquad
\Delta v_{\rm triangle}=-\frac12(160)(0.10)=-8\,\mathrm{m\,s^{-1}}.
$$
They have the same average acceleration but different peaks and different time histories.
\Image[/crash-pulse-equal-velocity-change.png]
{title=Ideal crash pulses with equal velocity change and different peak deceleration}
{description=Own synthetic stopping examples, not measured crash-test data. The vertical axis shows the positive deceleration magnitude, so its integral is speed lost. Time labels are milliseconds; the calculation uses seconds.}
For a constant $100\,\mathrm{kg}$ mass following either prescribed history, <Newton's second law> gives the same total <impulse>, $m\Delta v=-800\,\mathrm{N\,s}$, while the peak net <force> magnitude is $8000\,\mathrm N$ for the constant pulse and $16000\,\mathrm N$ for the triangular pulse. Equal pulse area therefore does not imply equal peak loading. Component motion can also differ from the motion prescribed at its support because of <mechanical compliance> and dynamic response.
As a historical apparatus example, the NASA study analyzed twelve light-aircraft crash tests and conducted two vertical drop tests with an energy-absorbing seat prototype at $12.8\,\mathrm{m\,s^{-1}}$ and $36g_n$ peak deceleration, where $g_n$ denotes <standard gravity>. Those test conditions are distinct from the synthetic example above.<Determination of crash test pulses and their application to aircraft seat analysis by E. Alfaro-Bou and others>[Author abstract: twelve crash tests and two vertical drop tests]{ref}
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