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 , can be broken down into the horizontal and vertical components , 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.