A car and a truck collide in an intersection, the former presented with a flashing red light and the latter an flashing yellow light. To determine which driver was at fault for the crash, it is best to work backwards through the scenario. Both vehicle skidded off at different angles. Because the vehicles' tires are locked after the crash, both vehicles will decelerate at a constant rate, eventually stopping at their final positions using this force of friction, the vehicles' mass at the distance traveled we can calculate the velocity of each vehicle using Newton's 2nd Law and kinematics. Because the vehicles' skidded off at an angle, their velocities have both X and Y components, with the truck traveling in the positive X direction and the car traveling in the positive Y direction.
An object's momentum is the product of its mass and its velocity, and is sometimes referred to as its tendency to stay in motion. Because both vehicles skidded in both X and Y directions after the collision, so they had momentum in both X and Y directions right after the vehicles hit each other. However, before the collision, the car only had momentum in the Y direction and the truck in the X direction, as that was the direction in which they were traveling. Momentum is conserved in any collision, meaning that the total amount of momentum in the X and Y directions will remain the same before and after the collision. This means that the momentum of the car right before the collision will be equal to the Y component of both vehicles' momentum after the collision, and the momentum of the truck right before the collision will be equal to the X component of both vehicles' momentum after the collision. From the momentum we can calculate the speed of each vehicle.
Lincoln Hawk, the truck driver, claims to have been traveling only 6.7m/s during the impact. The calculations will reveal whether he is telling the truth or not. Mike Rokar, the car driver, claims that he had come to a complete stop at the flashing red light before preceding into the intersection. Using kinematics, we can calculate the maximum speed the car could have reached from a stop in the distance from the light to the point of collision using the vehicle's maximum acceleration. If Mr. Rokar's car could have not reached the calculated speed before the collision, he must not be telling the truth.
To determine the acceleration the vehicles experience when skidding and therefore the velocity of the vehicles, we first must determine the coefficient of friction between each vehicle's tires and the road. According to police department findings, it takes 100N of force (23lb) to drag a 29lb (130N) tire. With this information we can calculate the coefficient of friction for the car:
Frictional Force = Normal Force • Coefficient of Friction
100N = 130N • μ
100N/130N = μ
μ = 0.769
According to the truck manufacturer, the coefficient of the truck's tires is only 70% of the car's, so the truck's coefficient of friction is:
μ = 0.538
100N = 130N • μ
100N/130N = μ
μ = 0.769
According to the truck manufacturer, the coefficient of the truck's tires is only 70% of the car's, so the truck's coefficient of friction is:
μ = 0.538
From these values, we can calculate the car and the truck exert a 10462N and 37531N frictional force on the ground respectively.
Force = Mass • Acceleration
Car: -10462N = 1387.7kg •a
a = -7.539m/s/s
Truck: -37530N = 7112.2kg • a
a = -5.277m/s/s
Car: -10462N = 1387.7kg •a
a = -7.539m/s/s
Truck: -37530N = 7112.2kg • a
a = -5.277m/s/s
We can then calculate the velocity of each vehicle right after the impact using kinematics. The car skidded 8.2m and the truck 11.0m before coming to a stop.
(final velocity)^2 = (initial velocity)^2 + 2 • (acceleration) • (change in position)
Car: (0m/s)^2 = v^2 + 2•-7.539m/s•8.2m
v = 11.119m/s
Truck: (0m/s)^2 = v^2 + 2•-5.277m/s•11.0m
v = 10.775m/s
Car: (0m/s)^2 = v^2 + 2•-7.539m/s•8.2m
v = 11.119m/s
Truck: (0m/s)^2 = v^2 + 2•-5.277m/s•11.0m
v = 10.775m/s
However, these velocities were at specific angles and we need to use trigonometry to calculate the X and Y component of each vehicle's velocity.
Car:
Y = 11.119sin(33) = 6.0558m/s
X = 11.119cos(33) = 9.3252m/s
Truck:
Y = 10.775sin(7) = 1.1313m/s
X = 10.775cos(7) = 10.694m/s
Y = 11.119sin(33) = 6.0558m/s
X = 11.119cos(33) = 9.3252m/s
Truck:
Y = 10.775sin(7) = 1.1313m/s
X = 10.775cos(7) = 10.694m/s
From these velocities we can calculate the component momentum of each vehicle:
Momentum = Mass • Velocity
Car:
Y = 6.0558m/s • 1387.7kg = 8403.6kg•m/s
X = 9.3252m/s • 1387.7kg = 12941kg•m/s
Truck:
Y = 1.1313m/s • 7112.2kg = 8046.0kg•m/s
X = 10.694m/s • 7112.2kg = 76057kg•m/s
Car:
Y = 6.0558m/s • 1387.7kg = 8403.6kg•m/s
X = 9.3252m/s • 1387.7kg = 12941kg•m/s
Truck:
Y = 1.1313m/s • 7112.2kg = 8046.0kg•m/s
X = 10.694m/s • 7112.2kg = 76057kg•m/s
While both cars have X and Y components of their momentum, the car only had momentum in the Y direction and the truck in the X direction before the crash. because momentum is always conserved in any collision, the sum of the momentum in the Y direction will be the car's momentum, while the sum of of the momentum in the X direction will be the trucks momentum.
Car:
8403.6kg•m/s + 8046.0kg•m/s = 16450kg•m/s
Truck:
12941kg•m/s + 76057kg•m/s = 88998kg•m/s
8403.6kg•m/s + 8046.0kg•m/s = 16450kg•m/s
Truck:
12941kg•m/s + 76057kg•m/s = 88998kg•m/s
From these values we can calculate the velocity of each vehicle just before the collision with their masses.
Car:
16450kg•m/s = 1387.7kg • v
v = 11.854m/s
Truck:
88998kg•m/s = 7112.2kg • v
v = 12.513m/s
16450kg•m/s = 1387.7kg • v
v = 11.854m/s
Truck:
88998kg•m/s = 7112.2kg • v
v = 12.513m/s
From this it is clear that Mr. Hawk was not telling the truth, as he was traveling nearly twice his reported speed at the time of the impact. The question still remains whether Mr. Rokar's car could have accelerated to a speed of 11.854m/s from a stop in his Ford Escort. According to Ford, the car has a maximum acceleration of 3.0m/s/s. We can calculate the minimum velocity of the car at the light for it to accelerate to 11.854m/s within the distance from the light to the collision zone (13.0m) using kinematics.
(final velocity)^2 = (initial velocity)^2 + 2 • (acceleration) • (change in position)
(11.854m/s)^2 = v^2 + 2•(3.0m/s)•(13m)
v = 7.9068 m/s
(11.854m/s)^2 = v^2 + 2•(3.0m/s)•(13m)
v = 7.9068 m/s
Mr. Rokar was traveling at the slowest, 7.9068m/s at the light, not making nearly a complete stop. Neither driver was truthful in their confession and both are at fault.