The Physics Of Collision

By: Gary E. Kilpatrick And Associates, PA
Forensic Engineers

Tel: (336) 841-6354
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Website: www.gekandassociates.com/

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When we drive our vehicles down the road, we are driving them at a certain velocity and direction. All vehicles have weight and mass to the order of a few hundred pounds for a motorcycle, a few thousand pounds for a large SUV or luxury car or several tons for a tractor-trailer truck. When our vehicles move at speed down the road, they create energy by virtue of their mass and forward velocity. This energy is called Kinetic Energy mathematically discribed as KE = 1/2mv² where m = mass and v = velocity. Assuming the mass of a vehicle is constant, it will create a certain quantity of kinetic energy while in motion at speed. When we accelerate our vehicles to a higher velocity, the vehicle's kinetic energy increases by the square of the velocity divided by 2. For example, if we consider a vehicle that weighs approximately 4000 pounds that is traveling at 60 mph which equals 88 feet per second (ft/s), this vehicle will produce approximately 480,994 ft•lb_f of kinetic energy. If the velocity of this vehicle is increased to 70 mph which is a 10 mph increase and a 16.7% increase in speed, the vehicle's kinetic energy increases to 654,686 ft•lb_f which is an increase of approximately 36.1%.

The physics of a collision between two bodies is governed by Newton's Second Law of Motion F = ma/g_c where F = force in pounds or lb_f, m = mass in pound mass or lb_m, a = acceleration in feet per second squared or ft/s² and g_c = a proportionality constant of 32.2(lbm•ft/lbf•s²). When vehicles are moving at fairly high rates of speed, they possess much kinetic energy. When they collide, the time of impact is very brief called an impulse. An example is when we watch a NASCAR race on TV, and a race car collides with a wall or another race car at say 150 mph. The forces generated due to these very brief impact time intervals are called impulsive forces. Newton's Second Law of motion can be restated as F = ma/g_c = mv/tg_c where v = velocity in feet per second or ft/s, t = time in seconds or s and v/t = the acceleration in ft/s². Due to this very short elasped time interval during a collision, it can be seen mathematically from this equation that this short elasped time interval causes the contact forces between two vehicle bodies to be very high. This is due to the time variable "t" being in the denominator of the equation. Forces can be generated at impact to the order of tens of thousands to hundreds of thousands of pounds depending on the mass of the vehicles and their velocity at impact.

For example, if we have two identical vehicles that weigh 4000 pounds each and they collide in a head on collision in a direct central impact where each vehicle is moving at say 55 mph, the combined closing velocity of the two vehicles at impact is 110 mph or 161.33 ft/s. When they collide with each other head on, both vehicle bodies absorb kinetic energy, collapse and crush inward. Assume the frontal crush measurement of each vehicle to be approximately five feet where the total crush of both vehicles is 10 feet. The average closing velocity of both vehicles is approximately 161.33 ft/s/2 = 80.66 ft/s. The elapsed time of the impact between the two vehicles is calculated using the distance formula d = vt where we solve this equation for the time "t" variable t = d/v and substituting the numbers we get 10ft/80.66ft/s = 0.123s or 123 milliseconds. Using these assumptions, the average force generated during the collision on each vehicle is approximately F = mv/tg_c = (4000lbm)(80.33ft/s)/(((32.2(lbm•ft/lbf•s))(0.123s)) = 81,129 pounds. Based on this calculation, we can see why a motor vehicle's body structure experiences the damage it incurs during a severe high speed collision.

Newton's Law above can be rearranged into the form of ∫f•dt = ∫m(dv/dt)dt. We can then apply integral calculus and integrate both sides of this equation to obtain f(t₂ - t₁) = mv₂ - mv₁ for a single mass. The term (t₂ - t₁) in this resulting equation is the elapsed time of impact for a mass, and the term mv is called the linear momentum of the mass where Newton calls this term the quanity of motion. The term (mv₂ - mv₁) equals the change in linear momentum of this mass during a collision. The term ∫f•dt is called the impulse of the collision.

When two vehicles collide with each other, the force that Vehicle A generates against Vehicle B is equal in absolute value to the force that Vehicle B generates against Vehicle A during the impact. These forces are equal in magnitude but are opposite in direction. There is a theoretical and instantaneous point during the impact between two vehicles where there is no relative motion between the two vehicles which is the point of maximum force and body compression. Assuming this, we can develop another equation utilizing the impulse ∫f•dt for each vehicle and set them equal to each other where one side of the equation must have a negative sign to indicate that the impulse of Vehicle A is opposite in direction to Vehicle B or ∫f_A•dt = -∫f_B•dt or (m_A•v_A2 - m_A•v_A1) = (m_B•v_B1 - m_B•v_B2). This equation can then be rearranged algebraically to obtain (m_A•v_A1 + m_B•v_B1) = (m_A•v_A2 + m_B•v_B2). This new equation is called the Conservation Law of Linear Momentum and is used to calculate the before impact velocities of two vehicles that collide with one another. Linear Momentum is always conserved in a collision because gravitational forces, frictional forces from skiding tire and air drag on the vehicle bodies are very small in comparision to the very high contact forces generated during a collision, and these much smaller forces can be assumed insignificant. Looking at the Conservation Law of Linear Momentum equation, it states that the combined momentum of both vehicles before a collision is equal in magnitude to the combined momentum of both vehicles after the collision. Since traffic accidents are bidirectional in nature, this equation is utilized in both the x and y directions by incorporating a branch of physics known as vector mechanics and a branch of mathematics called trigonometry which is the study of triangles, vectors, vector magnitudes, vector angles and their components. Because of the bidirectional nature of a collision between two vehicles, two equations one for the x direction and one for the y direction are created that generate a system of two equations with four unknown variables. These variables consist of the before impact velocities of the two vehicles in a collision and the after impact velocities for the same two vehicles in the collision. Once the direction vectors of both vehicles are measured and determined before and after the collision takes place at the accident scene and the after impact velocities of both vehicles are calculated as discribed below, then these two x and y Conservation Law of Linear Momentum equations can then be solved simultaneously to obtain the before impact velocities of the two vehicles.

The kinetic energy equation KE = 1/2mv² is used to obtain the after impact velocity of the vehicles involved in a collision. This equation is established to equal the work energy performed by skidding tires and/or the vehicle's body sliding across various surfaces after impact. When a vehicle's tires and/or body slide over various surfaces, the vehicle's kinetic energy is transformed to heat energy by way of frictional heating of the vehicle's, tires and/or body panels from sliding across one or more surfaces. This type of energy conversion is not conserved because the heat energy is released into the atmosphere and is lost, and the frictional forces acting on the vehicle to slow and stop it are the primary forces involved. Air drag is considered to be insignificant and is not included in the calculations because the vehicles slow quickly after a collision. The kinetic energy equation has the form 1/2mv²₂ = 1/2mv²₁ + N•µ•d + 1/2Iω² where the term 1/2Iω² can be used in the equation if a vehicle turns many 360's about its center of mass after a collision which accounts for rotational energy where ω = rotational velocity in radians per second or rad/s and I = the vehicle mass moment of inertia. The equation of the frictional forces developed by a skidding tire or body panel on some surface is N•µ•d. N = the Normal Force or perpendicular force of vehicle's weight over the tire or body panel. µ = the dynamic coefficient of friction between a tire or body panel and the surface it is skidding on. d = the distance the vehicle slides. N is dependant on the angle or slope of the road or ground surface. If the road surface has a significant slope say a 30% grade, then the angle of the road surface would be approximately 16.7°. This affects the amount of perpendicular vehicle weight over a tire or body panel. If a 1000 pounds of vehicle weight acts over a tire on a level surface, then the perpendicular component of the vehicle's weight over the tire at a inclination of 16.7° is reduced to 958 pounds. This can be calculated utilizing vector mechanics and trigonometry.

The term Drag Factor is utilized in traffic accident reconstruction. This term is not mentioned in engineering mechanics or physics textbooks. Drag Factor (f) and Coefficient of Friction (µ) are not the same. Drag Factor is the deceleration coefficient for an entire vehicle. The Coefficient of Friction is the deceleration coefficient for a sliding tire. The Drag Factor and Coefficient of Friction are the same, if and only if, all four tire on a motor vehicle are locked and sliding on a level surface. The equation for a vehicle's Drag Factor is denoted as f = a/g and f = F/w. "a" is equal to the deceleration of the vehicle, and "g" is the acceleration due to the earth's gravity. "F" = drag forces to slow a vehicle. "w" = vehicle's weight. If a vehicle decelerates at the rate of say 0.60g's, then the vehicle's drag factor is 0.60. It is a calculated number that is utilized in the kinetic energy equation in place of µ and averages the frictional drag of all four tires over both axles and is calculated by the engineer to represent the overall frictional drag factor on the entire vehicle as it slides and yaws over various surfaces just after the collision takes place. The drag factor equation for an entire vehicle is f_RD = ((f_f - x_f(f_f - f_r))/((1 - z(f_f - f_r)). The resultant drag factor calculation is very important when a vehicle yaws and spins out of control after a collision. As a vehicle yaws after impact, the drag forces generated by each tire and axle can vary. When a vehicle slides from one surface to another, from the asphalt to a dirt/grass,gravel/sand shoulder, the drag forces vary for each tire as individual tires slide from the asphalt to a dirt, grass, gravel or sand shoulder. The vehicle's tire drag forces are maximum during a forward skid with all four wheels locked and is somewhat less in a sideways skid and is minimum when the vehicle is free rolling forward or backward during its movement in a yaw (assuming the wheels are not locked due to impact damage). If one or more wheels of an axle is locked after a collision, for example, then the locked wheel will create more drag on an axle than the free wheel and must be taken into consideration in calculating the vehicle's drag factor. If the vehicle is skidding on an inclined surface, then the drag factor on a grade is f_RD = (µ + G)/SRT(1 - G²) where "G" is the grade of the surface expressed as a decimal and carries a positive sign for an incline surface and a negative sign for a decline surface.

In some cases, the collision forces are so violent that a vehicle's body can be literally torn into two large pieces. In these cases, the direction vector and movement distances of each vehicle body half must be measured and determined. The after impact velocity must be calculated for each vehicle body half. The engineer will then sum the two velocity vectors for each vehicle body haif which allows the engineer to calculate the velocity vector of the center of mass of the vehicle which is equal to the velocity of the vehicle when it has been torn into two pieces.

Once the after impact velocities of both vehicles have been determined as stated above, these velocity numbers along with the vehicle masses can then be plugged into the x and y Conservation Law of Linear Mometum equation. This generates two expressions, one for the x direction and one for the y direction where the before impact velocities of both vehicles become the unknown variables. These two equations then become a system of two linear equations with constant coefficients and two unknowns. They can then be solved simultaneously to obtain the before impact vehicular velocities for each vehicles. This calculation must be checked algebraically to make sure that each side of the equation matches numerically. If the check shows that each side of the equation does not match, then the trajectory angles should be reviewed and adjusted. The drag factor or coefficient of fraction may have to be adjusted as well in the after impact velocity calculations.

One more equation can be developed and can be utilized with the Conservation Law of Linear Mometum equation. This equation is derived from the impulse definition ∫f•dt and the assumption of a direct central impact which means that both masses strike each other along their centers of mass. During the derivation, the collision is made up of two subintervals of time, the period of deformation and the period of restitution. The period of deformation refers to the time duration of a collision that starts from the initial contact of the two masses and ends at the instant of maximum body deformation or compression. During this subinterval of time, the impulse is definded as ∫D•dt. The subinterval of restitution is the time interval from maximum body compression to the instant where the two masses just separate and is definded as ∫R•dt. Both of these impluses are equal in absolute magnitude but are opposite in direction during the point of maximum body compression. If the bodies are perfectly elastic, they will bounce off of each other and reestablish their original shape during the subinterval of restitution. However, when two bodies do not reestablish their shapes, we say that plastic deformation has occurred. The ratio of the impulse during restitution to the impulse of deformation ∫R•dt/∫D•dt is equal to a number called the coefficient of restitution (COR) and is denoted by the greek letter ε. It can be shown algebraically that this equation equals ε = -(V_B2 - V_A2)/(V_B1 - V_A1) and defines the relative velocity of separation to the relative velocity of approach. The COR for an impact between two vehicles is between 0 and 1. For an ideal and perfectly elastic collision where both bodies rebound and regain their initial shape before the impact took place much like two ceramic marbles, the COR = 1. For a totally plastic collision where two bodies would collide and stick together like soft putty, the COR = 0. The COR for two modern passenger vehicles that collide with each other depend on the speed and orientation at which they impact each other. According to Randall Noon in his 1994 book "Engineering Analysis of Vehicular Accidents", laboratory research has shown through fixed barrier impact studies that the COR can range from 0.20 with a speed of 25 mph, 0.1 with a speed of 35 mph and 0.004 with speeds of over 50 mph. Unless impacts studies of an actual collision are performed in a laboratory, which are not practical due to the cost involved in actual traffic accident reconstruction cases, then the COR will not be known exactly and can only be estimated and based on prior laboratory tests and texts.