Analysis of Collisions, Conservation of Linear Momentum: Can we do better?Raymond Brach, University of Notre Dame & Brach Engineering, LLCMatthew Brach, Brach Engineering, LLC

An article was published in the last issue ofCollision, “Analysis of Collisions, Point MassMechanics and Planar Impact Mechanics”, Vol 2, Issue1. The article summarized some of the main conceptsand equations of two dimensional point mass mechanics.Typically referred to as Conservation of LinearMomentum (COLM), the equations commonly are usedto reconstruct collisions. The use of COLM doesn’trequire the value of thecoefficient of restitution or thePDOF as input. Rather, thesevalues can be computed as partof the results and, particularly the coefficient of restitution,should be used to assess therealism of solutions. An example showed that if thevalue and sign of the coefficient of restitution is ignoredby the reconstructionist, the result could be a faultyreconstruction. Another important point made is that theuse of COLM ignores vehicle rotations. It was shownthat ignoring rotations can lead to significant errors inthe collision kinetic energy loss, the value and directionof each vehicle’s ΔV and each vehicle’s principaldirection of force (PDOF).

After covering point mass mechanics, the articleshifted its perspective and presented a completecoverage of Planar Impact Mechanics. In contrast toCOLM, planar impact mechanics is a more generalmethod of analyzing and reconstructing collisions in twodimensions. Moreover, the method takes angularmomentum of the vehicles into account. Despite theadditional complexity of taking the vehicles’ dimensionsand rotational velocity changes into account, the initialand final velocity components of both colliding vehiclescan be computed without a need for numerical solutions.Unfortunately, the solution equations in the articlecontained typographical errors. They should haveappeared as: (26)1 1 1(1 ) /n n rnV v m e v q m= + + (27)1 1 1(1 ) /t t rnV v m e v q mμ= + + (28)2 2 2(1 ) /n n rnV v m e v q m= − + (29)2 2 2(1 ) /t t rnV v m e v q mμ= − + (30)1 1 1(1 ) ( ) /rn c dm e v d d q Iω μΩ = + + −

(31)2 2 2(1 ) ( ) /rn a bm e v d d q Iω μΩ = + + −The velocity components in capital letters (V1n, V1t, V2n,V2t, Ω1 and Ω2) are final velocities; small, or lower caseletters, (v1n, v1t, v2n, v2t, ω1 and ω2) are initial velocities,

n and t refer to the normal and tangential coordinatesshown in Figure 6. A full set of equations for the planarimpact mechanics problem including definitions of all ofthe related variables such as vrn, q, μ and in Equationsm26 through 31 is presented in the original article.

Examples were presented in the article of howdifferences in the collision configurations of a crash(ignored when using COLM) can affect final angular

velocities. The examples wenton to show how thosedifferences significantly affectthe collision kinetic energy loss,the value and direction of eachvehicle’s ΔV and each vehicle’s

PDOF. These can be very important in anyreconstruction, but can play a critical role whenmatching a reconstruction to the speed change from anEvent Data Recorder (EDR), ΔVEDR. For a vehicle withan axial sensor, the resultant velocity change of thevehicle, ΔVVEH, (calculated using planar impactmechanics) is related to the value recorded by the EDRby (33)cos( )EDR VEHV V PDOFΔ = ΔAny errors in computing the ΔV and PDOF of a vehicleby ignoring rotations, could make a reconstruction usingEDR data invalid.

Main conclusions of the article included:P calculations using planar impact mechanics are morerigorous and accurate that those of COLM,P planar impact mechanics should always be used toreconstruct a collision rather than COLM,P reconstructions using planar impact mechanics areeasily computed using ordinary spreadsheet methods,MathCAD, MATLAB, etc. using Eq 26 - 31.

When is it best to use COLM?

Never.

y

Γ

x

m , I2 2

yxP

xPθ2

Py

C C

2ϕ

P

m , I1 1

1ϕ

θ1

tn

2d

d1

Figure 6. Free body diagrams of two colliding vehiclewith coordinate systems and variables.