Question 14.CS.1: Dynamics principles are used extensively in accident reconst......
Dynamics principles are used extensively in accident reconstruction. Police investigators measure skidmarks on pavement to determine the direction of motion, as well as to help determine the initial speeds of cars involved in an accident. Multiple car accidents can be especially difficult to analyze—a simplified case, shown in CS Fig. 14.1, is presented here.
Car A is traveling with a speed of (v_{A})_{0} when it hits car B, which is slowing down to a speed of (v_{B})_{0} because of stationary vehicles in front of it. For this preliminary investigation, assume that the two cars stick together, skid a distance d_{1}, and strike car C. The three vehicles stick together, then slide an additional distance d_{2} before coming to rest. By measuring the length of the skid marks, it is known that d_{1} = 30 ft and d_{2} = 20 ft.
The driver of car B claims he was traveling 20 mph when he was hit from behind. You have been asked to do a preliminary analysis to determine if the driver of car A was speeding, given that the speed limit at the site of the accident was 40 mph.
STRATEGY: In accident reconstruction, you typically work “backward.” When the cars are skidding, you can use work and energy to relate skid distances and vehicle velocities. You can use impulse and momentum to relate the velocities of the vehicles immediately before and after a collision.
MODELING: You can model the cars as particles, and create your impulsemomentum diagrams for each collision, as shown in CS Fig. 14.2.
ANALYSIS: Defining position 4 as the point where the cars come to rest, applying work and energy gives you
\begin{array}{c c}{{T_{3}\,+\,U_{3\to4}=T_{4}\colon\qquad{\frac{1}{2}}(m_{A}+m_{B}+m_{C})\,\nu_{3}^{2}-F_{f}d_{2}=0}}\end{array}
Use CS Fig. 14.3 to help you determine the value for F_{f} . Summing forces in the y direction gives you F_{f}=\mu N=\mu({\mathfrak{m}}_{A}+m_{B}+m_{C})g (this would be different if the cars were on a slope). Solving for v_{3} gives you
\nu_{3}={\sqrt{\frac{2\mu(m_{A}+m_{B}+m_{C})g\,d_{2}}{m_{A}+m_{B}+m_{C}}}}={\sqrt{2\mu g\,d_{2}}} (1)
Defining your system as cars A and B combined and car C, you can use CS Fig 14.2 and apply the conservation of linear momentum for the second impact, thus giving
(m_{A}+m_{B})\nu_{2}=(m_{A}+m_{B}+m_{C})\nu_{3}
Solving for \nu_{2}, you get
\nu_{2}=\frac{(m_{A}+m_{B}+m_{C})}{(m_{A}+m_{B})}\,\nu_{3} (2)
Now you must determine how much energy is lost between position 1 (right after collision 1) and position 2 (right before collision 2). Using an approach analogous to Eq. (1) above, you get
T_{1}+U_{1\to2}=T_{2}:\qquad\frac{1}{2}(m_{A}+m_{B})\:\nu_{1}^{2}-F_{f}d_{1}=\frac{1}{2}(m_{A}+m_{B})\:\nu_{2}^{2}Solving for v_1 gives you the velocity of cars A and B right after their collision:
\nu_{1}=\sqrt{\nu_{2}^{2}+\frac{2\mu(m_{A}+m_{B})g\,d_{1}}{m_{A}+m_{B}}}=\sqrt{\nu_{2}^{2}+2\mu g\,d_{1}} (3)
Finally, you can use impulse and momentum to determine the speed of car A before the collision:
m_{A}(\nu_{A})_{0}+m_{B}(\nu_{B})_{0}=(m_{A}+m_{B})\nu_{1}
Solving for (\nu_{\mathrm{A}})_{0} gives you:
(\nu_{A})_{0}=\frac{(m_{A}+m_{B})(\nu_{1})-m_{B}(\nu_{B})_{0}}{m_{A}} (4)
Knowing the makes of the cars, we can estimate the weights as W_{A} = 4000 lb, W_{B} = 3000 lb, and W_{C} = 2700 lb. From road testing, the kinetic coefficient of friction between rubber tires and dry concrete is μ = 0.70. Using these data and the value for d_{1}~and~d_{2}, and substituting Eq. (1) into Eq. (2), the resulting equation into Eq. (3), and finally this resulting equation into Eq. (4) gives you (v_{A})_{0} = 75.2 ft/s = 51.3 mph, which is well above the speed limit.
REFLECT and THINK: Accident reconstruction experts rely on the principles of dynamics to help them determine what happened during an accident. In the current example, you had to rely on the speed estimate of the driver who was rear-ended. If this driver had been stationary [e.g., (v_{B})_{0} = 0], then car A would have been going 66 mph. For car A to be driving at the speed limit, then (v_{B})_{0} would have an initial speed of approximate 61 mph (or 20 miles over the speed limit). The investigator would most likely depend on other information as well, such as eyewitness accounts and damage done to the different automobiles.
