The story
A truck and a car are both going through an intersection when the collide and hit each other. Mr. Rokar, driving the car, had a red light while Mr. Hawk, driving the truck, had a flashing yellow. This collision caused the cars to skid off in different angles away from the initial site of impact. During this time the cars wheels locked so the force of friction from the road to the wheels caused a constant deceleration for each ca until they eventually stop.
- Mr. Rokar (driving the car) claims that he made a full stop at the light before entering the intersection and that Mr. Hawk did not slow down prior to the collision.
- Mr. Hawk (driving the truck) claims to have been braking before the collision and that Mr. Rokar did not stop at the flashing red light.
Physics Background
Momentum is an objects tendency to stay in motion and is the product of mass and velocity. Much like a projectile, momentum has an X and a Y component that are completely independent from each other. Before the collision, the truck only has momentum in the X direction and the car only has momentum in the Y direction but afterwards they both have X and Y momentums. The law of conservation of momentum states that momentum is conserved in any collision so the momentum after the sum of the X and Y component of momentum after the collision and the initial X and Y momentum for the truck and car will be equal.
My Calculations
To figure out the deceleration of the vehicles while they were skidding and eventually the velocity of these vehicles we need to find out the frictional force between the tires and the road. The police department says that it would take 100N of force(23lb) to drag a 29lb (130N) tire. With this we can calculate the force of friction from the tire to the road.
Force Friction = Normal Force * Coefficient of Friction
100N= 130μ
100/130 = μ
0.769 = μ
Since the coefficient of friction for the tires is only 70% of what the trucks coefficient of friction is then we have to multiple that by 0.7.
0.769*0.7 = 0.538 = μ
Using this it is easy to determine that the normal force of the car is 10462 and the normal force of the truck is 37531 because you multiply the original 13600N of the car by 0.769 and the original 69700 of the truck by 0.538 This calculation is able to give us enough information to figure out the deceleration of the vehicles after the car.
F =ma
Car Truck
-10462N = 1387.7 kg * a -37531 = 7112.2 kg * a
-10462N/1387.7 kg= a -37531/7112.2kg = a
a = -7.539 m/s^2 a = -5.277 m/s^2
Using this and the knowledge that the truck skidded 8.2 m and the car skidded 11 m after the impact we can determine the velocity of each vehicle right after the impact.
(Velocity Final)^2 = (Velocity Initial)^2 + (2*(acceleration)(change in position))
Car Truck
(0m/s) = v^2 + (2*-7.539m/s^2*8.2m) (0m/s) = v^2 + (2*-5.277m/s^2*11m)
v = 11.119m/s v = 10.775 m/s
This doesn't account for the angle that the cars were traveling at though because nothing can be that simple. Using some easy trig the X and Y components can be found.
Car Truck
Y= 11.119sin(33) = 6.0558 m/s Y= 10.775sin(7) = 1.1313 m/s
X= 11.119cos(33) = 9.3252 m/s X= 10.775cos(7) = 10.694 m/s
Using this we can now calculate both the X and the Y components of momentum for both the car and the truck.
Momentum = Mass*Velocity
Car Truck
Y= 6.0558m/s*1387.7kg = 8403.6 kg*m/s Y= 1.1313m/s*7112.2kg = 8046 kg*m/s
X= 9.3252m/s*1387.7kg = 12941 kg*m/s X= 10.694m/s*7112.2kg = 76057 kg*m/s
Both vehicles have X and Y components of momentum which is differing from what their original components with the car only having momentum in the Y direction and the truck only having momentum in the X direction. Momentum though, is conserved in every collision so when adding both of these momentums together it will be equal to the initial momentum of the entire system which will help us find out who is at fault.
The cars momentum is the momentum in the Y and the trucks is the momentum in the X
Car Truck
8403.6+8046 = 16450kg*m/s 12941+76057 = 88998kg*m/s
Now we can calculate the velocity of the vehicles just before the collision.
Car Truck
16450kg*m/s = 1387.7kg*v 88998kg*m/s = 7112.2kg*v
(16450kg*m/s)/1387.7kg = v (88998kg*m/s)/7112.2kg = v
v= 11.854m/s v = 12.513m/s
Using this information it is very obvious that Mr. Hawk is not telling the truth since he said that he was going 6.7m/s at the time of impact and is actually traveling at a speed almost twice of that at 12.513m/s. The other question is if the car could accelerate to a speed of 11.854m/s from a complete stop in his Ford Escort. Ford Motor company has confirmed that the maximum acceleration of the Ford Escort would be about 3 m/s^2. Using this we can find the minimum velocity of the car using kinematics.
(Final Velocity)^2 = (Initial Velocity)^2 + (2(acceleration)(change in position))
(11.854m/s)^2 = v^2 + 2(3m's *13m)
v= 7.9068 m/s
The slowest that Mr. Rokar was traveling at the light would have been 7.9068m/s which is obviously not a complete stop making both of the drivers confessions false and causing both of them to be at fault.
Force Friction = Normal Force * Coefficient of Friction
100N= 130μ
100/130 = μ
0.769 = μ
Since the coefficient of friction for the tires is only 70% of what the trucks coefficient of friction is then we have to multiple that by 0.7.
0.769*0.7 = 0.538 = μ
Using this it is easy to determine that the normal force of the car is 10462 and the normal force of the truck is 37531 because you multiply the original 13600N of the car by 0.769 and the original 69700 of the truck by 0.538 This calculation is able to give us enough information to figure out the deceleration of the vehicles after the car.
F =ma
Car Truck
-10462N = 1387.7 kg * a -37531 = 7112.2 kg * a
-10462N/1387.7 kg= a -37531/7112.2kg = a
a = -7.539 m/s^2 a = -5.277 m/s^2
Using this and the knowledge that the truck skidded 8.2 m and the car skidded 11 m after the impact we can determine the velocity of each vehicle right after the impact.
(Velocity Final)^2 = (Velocity Initial)^2 + (2*(acceleration)(change in position))
Car Truck
(0m/s) = v^2 + (2*-7.539m/s^2*8.2m) (0m/s) = v^2 + (2*-5.277m/s^2*11m)
v = 11.119m/s v = 10.775 m/s
This doesn't account for the angle that the cars were traveling at though because nothing can be that simple. Using some easy trig the X and Y components can be found.
Car Truck
Y= 11.119sin(33) = 6.0558 m/s Y= 10.775sin(7) = 1.1313 m/s
X= 11.119cos(33) = 9.3252 m/s X= 10.775cos(7) = 10.694 m/s
Using this we can now calculate both the X and the Y components of momentum for both the car and the truck.
Momentum = Mass*Velocity
Car Truck
Y= 6.0558m/s*1387.7kg = 8403.6 kg*m/s Y= 1.1313m/s*7112.2kg = 8046 kg*m/s
X= 9.3252m/s*1387.7kg = 12941 kg*m/s X= 10.694m/s*7112.2kg = 76057 kg*m/s
Both vehicles have X and Y components of momentum which is differing from what their original components with the car only having momentum in the Y direction and the truck only having momentum in the X direction. Momentum though, is conserved in every collision so when adding both of these momentums together it will be equal to the initial momentum of the entire system which will help us find out who is at fault.
The cars momentum is the momentum in the Y and the trucks is the momentum in the X
Car Truck
8403.6+8046 = 16450kg*m/s 12941+76057 = 88998kg*m/s
Now we can calculate the velocity of the vehicles just before the collision.
Car Truck
16450kg*m/s = 1387.7kg*v 88998kg*m/s = 7112.2kg*v
(16450kg*m/s)/1387.7kg = v (88998kg*m/s)/7112.2kg = v
v= 11.854m/s v = 12.513m/s
Using this information it is very obvious that Mr. Hawk is not telling the truth since he said that he was going 6.7m/s at the time of impact and is actually traveling at a speed almost twice of that at 12.513m/s. The other question is if the car could accelerate to a speed of 11.854m/s from a complete stop in his Ford Escort. Ford Motor company has confirmed that the maximum acceleration of the Ford Escort would be about 3 m/s^2. Using this we can find the minimum velocity of the car using kinematics.
(Final Velocity)^2 = (Initial Velocity)^2 + (2(acceleration)(change in position))
(11.854m/s)^2 = v^2 + 2(3m's *13m)
v= 7.9068 m/s
The slowest that Mr. Rokar was traveling at the light would have been 7.9068m/s which is obviously not a complete stop making both of the drivers confessions false and causing both of them to be at fault.