Incident Details
The crash that I am investigating is between Mike Rokar, driving a compact car, and Lincoln Hawk, driving a tractor-trailer. The crash occurred in the middle of an intersection in which Mr. Rokar was approaching from a flashing red light, and Mr. Hawk from a flashing yellow light. Both drivers claim they aren't at fault: Mr. Rokar that he came to a complete stop at the intersection, and Mr. Hawk that he adequately slowed down at the intersection, at a speed of 6.7 m/s. To investigate the situation, I used principles of forces & motion, friction, momentum, and kinematics.
Crash Details
The following is the information I used to solve the dispute:
- It takes 100 newtons of force to drag a 13 kg (130 Newton) car tire across the pavement
- the coefficient of friction between the truck tires and the ground is 70% that of care tires
- After the collision the car and the truck skidded 8.2 meters at an angle of 33° and 11.0 meters at an angle of 7°, respectively
- The weight of the car is 1383 kg (13,600 Newtons) and the weight of the truck is 7119 kg (69,700 Newtons)
- The pre-crash angle of the car and the truck was 90°
- Mr. Hawk (the truck driver) claimed to have began braking in anticipation of a collision, traveling only at a speed of 6.7 m/s at the moment of impact
- Measurements show that the distance from the traffic light to the spot of impact of the impact for the car is 13 m
- Ford Motor Corporation specifications indicate that the maximum acceleration of a similarly loaded Ford Escort of approximately 3.0 m/s/s
Process & Calculations
To solve this dispute, I worked backwards, finding the momentum/velocities of the car & truck after the collision then using the properties of motion and momentum to solve for the velocities before the collisions, and used that information to determine who's lying.
Firstly, I determined the coefficient of friction, μ, between the car tires and the road. We're given that it takes 100N of force to drag a 13 kg (130 N) car tire across the pavement, so we can use the friction equation to get that:
Ff [car] = Fn [car] * μ [car] 100 N = 130 N * μ [car] μ [car] = 0.769
Using the information we have about the trucks tires relative to car's tires we can determine μ [truck]:
μ[truck] = μ[car] * 0.7 μ [truck] = 0.769 * 0.7 μ [truck] = 0.538
Using these values I determined the force of friction between the car and the truck and the ground
Ff [car] = Fn [car] * μ[car] Ff [truck] = Fn [truck] * μ[truck]
Ff [car] = 13600 N * 0.769 Ff [truck] = 69700 N* 0.538
Ff[car] = 10458.4 N Ff[truck] = 37498.6 N
Then we can calculate the acceleration of the car and truck would have as they're skidding to a stop
F[car] = m[car] * a[car] F[truck] = m[truck] * a[truck]
-10458.4 N = 1383 kg * a [car] -37498.6 N = 7119 kg* a [truck]
a [car] = -7.56 m/s/s a [truck] = -5.27m/s/s
Using Kinematics, I found the velocity directly after impact (va = velocity after crash)
(vf [car])^2 = (va [car])^2 + 2(a [car] * Δx) (vf [truck])^2 = (va [truck])^2 + 2(a [truck] * Δx)
0 = (va [car])^2 + 2 (-7.56 m/s/s* 8.2 m) 0 = (va [truck])^2 + 2 (-5.27 m/s/s* 11.0 m)
va [car] = 11.13 m/s va [truck] = 10.77 m/s
Since the car and truck both traveled at and angle after the collision, these velocities have X and Y components to them, and using the principles of Trigonometry and Forces at Angles I determined these components
vx [car] = cos(7°) * 11.13 m/s vx [truck] = cos(33°) * 10.77 m/s
vx [car] = 9.50 m/s vx [truck] = 10.69m/s
vy [car] = sin (7°) * 11.13 m/s vy [truck] = sin (33°) * 10.77 m/s
vy [car] = 6.17 m/s vy [truck] = 1.31 m/s
Now that we have the X and Y components of velocity after the collision, I used these, along with the masses of the car and truck, to find the X and Y components of momentum of the truck and car. I do this because of the law of linear momentum: momentum is conserved during a collision. Since the truck was traveling entirely in the X direction and the car entirely in the Y direction, summing the X and Y components of momentum, respectively, I found the momentum of the both the car and the truck before the crash.
px [car] = m [car] * vx [car] px [truck] = m [truck] * vx [truck]
px [car] = 1383 kg * 9.50 m/s px [truck] = 7119 kg * 10.69 m/s
px [car] = 13138.5 kg m/s px [truck] = 76102 kg m/s
py [car] = m [car] * vy [car] py [truck] = m [truck] * vy [truck]
py [car] = 1383 kg * 6.17 m/s py [truck] = 7119 kg * 1.31 m/s
py [car] = 8533.11 kg m/s py [truck] = 8044.47 kg m/s
Summing the X & Y components of momentum separately I used the conservation of linear momentum to find the momentum of the truck and car, respectively, before the collision (pb = momentum before collision):
pb [car] = py [car] + py [truck] pb [truck] = px [car] + px [truck]
pb [car] = 8533.11 kg m/s + 8044.47 kg m/s pb [truck] = 13138.5 kg m/s + 76102.11 kg m/s
pb [car] = 16577.58 kg m/s pb [truck] = 89240.61 kg m/s
Now that we have the momentum before the collision of both vehicles, we can use the momentum equation to figure out the velocities before the crash, and use that information to determine who is lying about the incident, and figure out who is at fault for the crash (vb = velocity before the crash).
pb [car] = m [car] * vb [car] pb [truck] = m [truck] * vb [truck]
16577.58 kg m/s = 1383 kg * vb [car] 89240.61 kg m/s = 7119 kg * vb [truck]
vb [car] = 11.99 m/s vb [truck] = 12.54 m/s
With this information it is evident that Mr. Hawk is lying since he claimed to be only at a speed of 6.7 m/s, when in reality his actual speed is about double that, meaning that he is at least partially at fault for the crash.
To determine if Mr. Rokar is lying I used kinematics to determine if it is possible to reach a speed of 11.99 m/s with a maximum acceleration of 3.0 m/s/s and a distance of 13 meters to travel. In other words, using kinematics I determined the lowest initial velocity that the car could be traveling at to reach the crash location at a speed of 11.99 m/s, and if that number is significantly greater than 0, Mr. Rokar is lying and he is at fault too (vl = minimum velocity of the car at the stop light)
(vb [car])^2 = (vl [car])^2 + 2 (a [car] * Δx)
(11.99)^2 = (vl [car])^2 + 2 (3.0 * 13)
vl [car] = 8.11 m/s
Since this number if significantly greater than 0, Mr. Rokar lied about stopping at the flashing red light, and is therefore at fault for the collision.
Both drivers are partially at fault for the collision, who's more at fault is up to the courts to decide given the information, but since Mr. Rokar was suppose to stop at a flashing red light and didn't come close, while Mr. Hawk was suppose to just slow down, in my opinion Mr. Rokar, the car driver, is more at fault.
Ff [car] = Fn [car] * μ [car] 100 N = 130 N * μ [car] μ [car] = 0.769
Using the information we have about the trucks tires relative to car's tires we can determine μ [truck]:
μ[truck] = μ[car] * 0.7 μ [truck] = 0.769 * 0.7 μ [truck] = 0.538
Using these values I determined the force of friction between the car and the truck and the ground
Ff [car] = Fn [car] * μ[car] Ff [truck] = Fn [truck] * μ[truck]
Ff [car] = 13600 N * 0.769 Ff [truck] = 69700 N* 0.538
Ff[car] = 10458.4 N Ff[truck] = 37498.6 N
Then we can calculate the acceleration of the car and truck would have as they're skidding to a stop
F[car] = m[car] * a[car] F[truck] = m[truck] * a[truck]
-10458.4 N = 1383 kg * a [car] -37498.6 N = 7119 kg* a [truck]
a [car] = -7.56 m/s/s a [truck] = -5.27m/s/s
Using Kinematics, I found the velocity directly after impact (va = velocity after crash)
(vf [car])^2 = (va [car])^2 + 2(a [car] * Δx) (vf [truck])^2 = (va [truck])^2 + 2(a [truck] * Δx)
0 = (va [car])^2 + 2 (-7.56 m/s/s* 8.2 m) 0 = (va [truck])^2 + 2 (-5.27 m/s/s* 11.0 m)
va [car] = 11.13 m/s va [truck] = 10.77 m/s
Since the car and truck both traveled at and angle after the collision, these velocities have X and Y components to them, and using the principles of Trigonometry and Forces at Angles I determined these components
vx [car] = cos(7°) * 11.13 m/s vx [truck] = cos(33°) * 10.77 m/s
vx [car] = 9.50 m/s vx [truck] = 10.69m/s
vy [car] = sin (7°) * 11.13 m/s vy [truck] = sin (33°) * 10.77 m/s
vy [car] = 6.17 m/s vy [truck] = 1.31 m/s
Now that we have the X and Y components of velocity after the collision, I used these, along with the masses of the car and truck, to find the X and Y components of momentum of the truck and car. I do this because of the law of linear momentum: momentum is conserved during a collision. Since the truck was traveling entirely in the X direction and the car entirely in the Y direction, summing the X and Y components of momentum, respectively, I found the momentum of the both the car and the truck before the crash.
px [car] = m [car] * vx [car] px [truck] = m [truck] * vx [truck]
px [car] = 1383 kg * 9.50 m/s px [truck] = 7119 kg * 10.69 m/s
px [car] = 13138.5 kg m/s px [truck] = 76102 kg m/s
py [car] = m [car] * vy [car] py [truck] = m [truck] * vy [truck]
py [car] = 1383 kg * 6.17 m/s py [truck] = 7119 kg * 1.31 m/s
py [car] = 8533.11 kg m/s py [truck] = 8044.47 kg m/s
Summing the X & Y components of momentum separately I used the conservation of linear momentum to find the momentum of the truck and car, respectively, before the collision (pb = momentum before collision):
pb [car] = py [car] + py [truck] pb [truck] = px [car] + px [truck]
pb [car] = 8533.11 kg m/s + 8044.47 kg m/s pb [truck] = 13138.5 kg m/s + 76102.11 kg m/s
pb [car] = 16577.58 kg m/s pb [truck] = 89240.61 kg m/s
Now that we have the momentum before the collision of both vehicles, we can use the momentum equation to figure out the velocities before the crash, and use that information to determine who is lying about the incident, and figure out who is at fault for the crash (vb = velocity before the crash).
pb [car] = m [car] * vb [car] pb [truck] = m [truck] * vb [truck]
16577.58 kg m/s = 1383 kg * vb [car] 89240.61 kg m/s = 7119 kg * vb [truck]
vb [car] = 11.99 m/s vb [truck] = 12.54 m/s
With this information it is evident that Mr. Hawk is lying since he claimed to be only at a speed of 6.7 m/s, when in reality his actual speed is about double that, meaning that he is at least partially at fault for the crash.
To determine if Mr. Rokar is lying I used kinematics to determine if it is possible to reach a speed of 11.99 m/s with a maximum acceleration of 3.0 m/s/s and a distance of 13 meters to travel. In other words, using kinematics I determined the lowest initial velocity that the car could be traveling at to reach the crash location at a speed of 11.99 m/s, and if that number is significantly greater than 0, Mr. Rokar is lying and he is at fault too (vl = minimum velocity of the car at the stop light)
(vb [car])^2 = (vl [car])^2 + 2 (a [car] * Δx)
(11.99)^2 = (vl [car])^2 + 2 (3.0 * 13)
vl [car] = 8.11 m/s
Since this number if significantly greater than 0, Mr. Rokar lied about stopping at the flashing red light, and is therefore at fault for the collision.
Both drivers are partially at fault for the collision, who's more at fault is up to the courts to decide given the information, but since Mr. Rokar was suppose to stop at a flashing red light and didn't come close, while Mr. Hawk was suppose to just slow down, in my opinion Mr. Rokar, the car driver, is more at fault.