# ENERGY, MOMENTUM AND DRIVING

When you are sitting in a moving vehicle you and the vehicle are subject to Newton's Laws of motion. Your vehicle's speed and direction and your body's speed and direction cannot change without external forces. The external force on the vehicle comes from another vehicle which is either moving or stationary, a stationary object, or gravity. The external forces can cause damage to the vehicle and injury to your body. The amount of damage or injury is determined by the magnitude of the force and by the part of the vehicle or the body the force is applied to. Minimizing damage to the vehicle and injury to the occupants often present conflicting requirements. Bumpers are designed to protect the car, airbags are designed to protect people. Neither of these does a perfect job. Both involve trade-offs. To learn more about air-bags visit the Insurance Institute For Hyway Safety where we got the air-bag picture.

### The Role of Momentum.

Newton's second law requires that the integral of force with respect to time must equal to the change in momentum. This implies that a given momentum change can be accomplished with a weaker forces if the time of interaction is increased. Give yourself more time to brake and the forces will be more gentle. Newton's third law requires that the forces acting on two interacting (colliding) objects will be equal and opposite. Your textbook shows how Newton's third law implies conservation of total linear momentum. The change in momentum of one colliding vehicle is accompanied by an equal and opposite change of momentum of the other vehicle. This idea is usually the underlying principle used in accident reconstruction.

### The Role of Mass And Energy.

When a light car and a massive truck collide, momentum conservation requires that

the change in Mtruckvtruck = the change in Mcarvcar

Note that an object with large mass will experience a smaller change in speed than the object with the small mass. As the occupant of the vehicle it is in your interest that the vehicle does not undergo big changes in speed in a short time. Since you yourself are a moving object subject to Newton's laws, you will need external forces yourself to change your motion. These will come from the seat, the seat belt, airbags or other parts of the vehicle such as the steering column.

If the vehicle comes to a stop, or if the speed is substantially decreased, the lost kinetic energy of the vehicle transforms into some other form. This process will involve work, the integral of force with respect to displacement. The bigger the displacement the weaker the force (and damage and injury.) Hard stationary objects don't allow much displacement and cause more damage and injury than soft, movable or breakaway objects. Next time you are driving or riding in car, look around and see how modern vehicle and highway designs surround you with soft, breakaway barriers whenever possible. Wide medians, breakaway lamp posts, energy absorbing barriers at underpasses, padded dashboards.

### Accident Reconstruction

Note: What follows is based on a contribution by Prof. David Wagner of Edinboro University. Dr. Wagner is a physics professor who is also a valued accident reconstruction expert.

Unfortunately, every day there are thousands of traffic accidents throughout the United States. While most accidents are minor, a significant number of accidents result in some form of legal action, either civil or criminal.

When legal action occurs, the courts get technical help from
(1) one of the many traffic institutes around the country, and
(2) from engineers and scientists whose specialty lends itself to some aspect of accident reconstruction.
Physicists fall into the second category.

It should be noted that it is rare for a reconstructionist to have an opportunity to be at a "hot" accident site. The reconstructionist usually has to rely on evidence gathered by the police or other investigators.

At the simplest level, a reconstruction might involve computing the location of a vehicle at various times before a collision. Or the problem might be to determine how far a vehicle traveled during the time when a driver was perceiving and reacting to the incident. Accident reconstructionist are paid up to several hundred dollars per hour for their time.

Of more interest are situations involving vehicles skidding, colliding, or yawing (sideslipping while traversing a curve). In these cases, the concepts of energy, momentum, and force are central.

### Skidding

When a vehicle begins to skid, the problem becomes one of classic kinetic friction. Given a knowledge of the coefficient of friction between the tires and the road, the slope of the road surface, and the total distance of the skid, it is quite straightforward to estimate the speed of the vehicle.

As an example, suppose that a vehicle is found to have skidded 130 feet along a flat road, coming to a complete stop. Given that the coefficient of friction of the tire/road surface is known to be between 0.55 and 0.70, what is the range of possible speeds?

 original kinetic energy = energy lost in sliding 1/2 mv2 = f mg d where d = distance of the skid, f = coefficient of friction Solving for v gives; v = (2fgd)1/2 Using the values from above yields for f = 0.55, v = 67.6 ft/s = 46.1 mph for f = 0.70, v = 76.3 ft/s = 52.0 mph
Thus it can be estimated that the driver was traveling between 46 and 52 mph at the time the brakes were applied.
With this speed known, and given typical driver reaction times, it is possible to paint a fairly complete scenario of the motion of the vehicle just prior to the accident.

## Collisions

Collisions pose a more severe challenge. In cases where skid marks completely document vehicle motions prior to and following a collision, the principle of conservation of momentum can be used to analyze the collision phase of an accident. For example, by means discussed above, a determination of the speed and direction of both vehicles immediately following impact (direction, of course, being determined by the line of the skid marks).

With the speed and direction of both vehicles known just after impact, conservation of momentum can be used to determine the speed and direction of both vehicles just prior to impact (again, skid marks specifying pre-impact direction).

Finally, skidding analysis applied to the pre-impact skid marks allows a determination of the original speeds of both vehicles.

### Energy and Collisions

During a collision, energy is also conserved. But the use of this principle requires that all the energy transformations occurring during the collision be amenable to analysis. The most difficult contribution to accurately determine is the energy of de post-impact skidding.
It turns out that tests with hundreds of vehicles have shown that there is a linear relationship between the amount of residual crush (crush remaining after the collision) and the energy lost to deformation and crush.
Energy of crush is a less precise reconstruction tool than is conservation of momentum. This is because, although there is an empirical linear relationship between crush and energy loss, individual vehicles only approximately adhere to this relationship

### Yaw Marks

When a vehicle is guided through a curve, even at constant speed, that vehicle is accelerating toward the center of the curve. The force that produces the acceleration is the side thrust between the tires and the road; that is to say, friction. Since f the yaw marks have crossways striations which are different from the marks left by a skidding tire.
There is a simple relationship between the radius of curvature of the yaw marks, the coefficient of friction, and the speed of the vehicle leaving the marks. The underlying physical principle is;

 Maximum sideways force of friction = vehicle mass x centripetal acceleration or fmg = mv2/r where r = radius of curvature of yaw mark Thus v = (fgr)1/2

Clearly, once it has been determined that a vehicle left yaw marks as a result of traversing a turn at the limit of friction, the speed of the vehicle can be determined.

Here are additional info links. We spend a lot of time and money on cars. It is worth taking an hour or two to examine some of the many interesting technical issues.

The final two links are not related to physics but you might find them interesting.
7. 8.

For extra credit answer the following questions.

1. What do traffic accident investigators use to estimate the speed and direction of travel before the accident.

2. Accident investigator manuals in this country define momentum as weight x velocity. This definition is obviously quite different from the physics definition of momentum as mass x velocity. Why is the former definition just as good for accident reconstruction purposes?