HALL - CARPENTER, CO.

By Edwin T. Scallon, Copyright © 1990, 1995, 2008 All Rights Reserved

ACCIDENT RECONSTRUCTION AND BAC CALCULATION PROGRAMS

CONSERVATION OF LINEAR MOMENTUM

    Conservation of Linear Momentum is not a new theory by any means, however, the formula for determining the exact speeds of both vehicles in a two car collision can be calculated at the same time.  The basic theory is that energy in a system can not be destroyed it must be converted. So the basis comes from Newton's three laws of motion. 

    In 1687 Isaac Newton first presented the three basic laws governing the motion of a particle. These laws gave insight into the effects of forces acting on bodies in motion. The three laws of motion are as follows: (1) Newton's first law of motion, the law of inertia, (2) Newton's second law of motion, the law of constant acceleration, (3) Newton's third law of motion, the law of momentum. Newton's First Law of Motion, the Law of Inertia A particle originally at rest, or moving in a straight line with a constant velocity, will remain in this state provided the particle is not subjected to an unbalanced force. The law of inertia describes the fundamental property of matter or a particle. Every object (body) remains in a state of rest or of uniform motion in a straight line unless acted upon by outside forces. This law states that motion is as natural a condition as rest. Just as an object at rest is in equilibrium, so is an object moving in a straight line at a constant speed in equilibrium.

    For example, a car that is going along a level road at constant speed is balanced by the supporting forces of the pavement. The forward pull of the engine counterbalances the retarding forces of friction and air resistance. The resultant force is zero; thus, the car is in equilibrium.

Newton's Second Law of Motion, The Law of Constant Acceleration: A particle acted upon by an unbalanced force F experiences an acceleration that has the same direction as the force and a magnitude that is directly proportional to the force. The law of constant acceleration describes what happens to a body when an external force is applied to a body. That body acted upon by a constant force will move with constant acceleration in the direction of the force; the amount of acceleration will be directly proportional to the acting force and inversely proportional to the mass of the body.

This law can be expressed by: F = ma or a = F/m

    Remember what happens when an object falls. It is accelerated by the force of gravity. In this example, the force is any applied force that is equal to the weight of the object, and the acceleration is that of gravity. Any weight unit can be used for F and W, and any acceleration unit (such as ft/sec/sec) can be used for a and g.

    For example, a car weighing 3200 lbs. accelerates at a rate of 5 ft/sec/sec. Ignoring friction, what is the effective forward force exerted by the engine? W equals 3200 lb. (the weight of the car); a equals 5 ft/sec/sec (the acceleration of the car) and g equals 32 ft/sec/sec (the acceleration of gravity). F=W a/g 500 lb. force = 3200 x 5/32 This law explains why pilots and astronauts experience G forces. The weight of a person at rest is the force exerted by the acceleration of gravity, 1 G, or 32 ft/sec/sec. If someone is being accelerated at a rate greater than 1 G, he will feel heavier.

    For example, if he is accelerated at 65 ft/sec/sec, he will feel as though his weight had doubled. An easy way to say this is that he is feeling a 2 G (2 x 32.2 = 64.4) force. If accelerated at 96 ft/sec/sec, he will feel a 3-G force, and so on.

Newton's Third Law of Motion, the Law of Momentum: The mutual forces of action and reaction between two particles are equal, opposite, and collinear. Often in a momentum analysis the weights of the vehicles are used in place of the vehicle's mass. The weight is not divided by the acceleration of gravity when the calculations are made. P = wv Because the acceleration of gravity is a constant, it can be seen that the momentum is directional proportional to an object's weight and velocity. That is, if you double the weight of an object and keep the velocity the same, the momentum is twice as much. Or if you double the velocity with the weight held constant, the momentum is doubled. Therefore, if you had a 3,000 lb car traveling eastbound at 40 ft/sec, it would have the same momentum as a 6,000 lb. vehicle traveling eastbound at 20 ft/sec.

                                                                                P=wv  P =120,000 ft - lbs. /sec

                                                                                P = 3000 lbs. (40 ft /sec)

                                                                                P=wv P = 6000 lbs. (20 ft /sec)

                                                                                P =120,000 ft - lbs. /sec 

    The law of conservation of momentum can be stated as: In any group of objects that act upon each other, the total momentum before the action equals the total momentum after the action. In traffic accident reconstruction, the action is the collision between two vehicles and the objects are the two vehicles. Using Newton's Second and Third Laws of Motion, an equation for conservation of momentum can be developed as follows:

v1 m1 + vm =  vm1 + vm2

The subscripts refer to vehicles one and two. The arrows above the velocity indicate that each term is a vector quantity. The symbol ' above the velocity is read "velocity prime" and refers to the aftercollision velocity. The left side of the equation is the momentum before the collision and the right side is the momentum after the collision.

    Because the equation has mass in each term and the mass is equal to the weight divided by the acceleration of gravity (m = w/g), w/g can replace the value for m in each term of the equation. Drop the g from each term and the equation becomes:

     v1 w1 + v2 w =  v1 w1 + v2 w2

    This is the equation used in making momentum calculations. It is a vector equation and is not a simple algebraic equation. The arrows over the terms are indicative of a vector equation. This equation states that the momentum before the collision is equal to the momentum after the collision.

    In practice, we need to establish the following data: 

    1.  The weight of the first vehicle W

    2.  The weight of the second vehicle W

    3.  The Point of Impact, (POI), and the angles of approach before initial engagement

    4.  The speed from skid marks for vehicle #1

    5.  The speed from skid marks for vehicle #2

    6.  The final resting plaace and distance from POI for vhicle #1

    7.  The final resting place and distance from POI for vehicle #2

    8.  Post impact speed from tire marks for vehicle #1

    9.  Post impact speed from tire marks for vehicle #2

   10. The angle of departure from POI for vehicle #1

   11. The angle of departure from POI for vehicle #2

   All the measurements for angles shall be calculated through the center of mass for each vehicle and shall be calculated using a diagram showing the x and y axis at 90º angles.  Once that is produced and the data is collected and recorded you are ready to calculate the speeds of both vehicles using the above formula.  An example of an accident I reconstructed in 1990 where one vehicle operated left of center on a rural road and was in a head on collision with the oncoming vehicle from the opposite directions.  Below is the diagram.

                              

  RETURN TO ACCIDENT RECON MENU

 RETURN TO HOME PAGE