Trigonometry in Automobile-Accident Reconstruction
by Linda Griffin Caples
in the Mathematics Teacher
January 1992
Trigonometry and vectors are used in helping to determine the speeds of automobiles in a car
accident. A quantity that plays a central role in accident reconstruction is the coefficient of
friction, f, which is defined by the equation
, where F is the
force (a vector) and W is the weight. When a driver slams on the brakes and the car skids to a
stop, its minimum speed can be estimated by using the formula
, where S is the
speed of the car in miles per hour, f is the drag factor or coefficient of friction, and d is the
length of the skid marks measured in feet.
How fast was a car going which skidded 70 feet on dry brick (coefficient of friction is .7 for
dry brick)? ______________
How fast was a car going which skidded 40 feet on wet oiled gravel (drag factor is .4 for wet
oiled gravel)? ______________
Check your answers with the nomograph from AAA.
Conservation of Momentum:
Linear momentum is the product of mass and velocity. By applying the law of conservation
of linear momentum, we can obtain a vector equation that will serve as a means of
determining unknown speeds for two vehicles in a collision. The total momentum before the
collision of the vehicles is equal to the total momentum after the collision. The corresponding
vector equation is
,
where M1 and M2 represent the masses of car 1 and car 2, respectively, with v1 and v2 the
corresponding velocities before impact and v3 and v4 the velocities after the collision.
We know that
, where M denotes mass, W denotes weight, and g denotes gravity (32
feet per second per second).
So, the equation above can be written as:
,
where W1 and W2 are the respective weights of the two vehicles involved. The resultant
vector (figure 1) obtained by adding
vertical components
, can be broken down into the horizontal and
, and ___________________________,
respectively, where
and
represent the directions of vectors v3 and v4. By using the
formula for the conservation of linear momentum and taking the magnitudes of vertical and
horizontal components, we obtain
,
and
,
with
and
being the directions for vectors v1 and v2.
Police officers make a scale drawing of the accident scene and measure skid marks in
accidents where loss of life occurs or where substantial damage occurs. An accident is
illustrated in figure 2. Let the x-axis be the path of approach to impact of car 1. We locate the
center of mass of each car as the point of intersection of the lines joining each front tire to its
diagonally opposite rear tire. For each car, the vector from center of mass at impact to the
center of mass at final resting position is drawn and measured to determine the distance and
angle.
In figure 4, we have these vectors,
, and
positioned at the origin and illustrating
the paths of the two vehicles after impact. These vectors are 25.5 feet and 16.75 feet,
respectively. If we take the drag factor at the time of impact to be f = 0.83, then
magnitude of v3 = _____________ and
magnitude of v4 = _____________ .
We note that the vectors v1, v2, v3, and v4 have directions, measured from the positive x-axis,
of 180 , 93 , 148 , and 154 , respectively. Letting W1 and W2 be 4,220 and 3,875 pounds,
respectively, and using the previous values for the magnitudes of vectors v3 and v4, substitute
into
.
We obtain
so the magnitude of vector v2 = ___________________.
,
Then substitute this value into
to obtain the magnitude of vector v1 = ___________________.
These are the speeds right before impact. By substituting into the Combined-Speed formula,
,
(where Sf is the final speed and S0 is the initial speed) we get the result that our drivers were
travelling at 48 mph and 37 mph the split second before they slammed on their brakes.