Background
Mike Rokar was driving a car and Lincoln Hawk was driving a truck and they crashed at the intersection of Furiosa Dr. and Fury Rd. At the intersection, the truck driver had a flashing yellow light and the car driver had a flashing red light. Each driver tells a different story of what happened, so I have to figure out which driver was at fault.
- Mike Rokar (car driver) 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
- Lincoln Hawk (truck driver) claims that he started to brake before the collision, and that Mr. Rokar didn't stop at the flashing red light.
Summary of My Calculations
First, I needed to find the coefficient of friction for tires on the road. The police department determined that the force required to drag a 130 N car tire across the pavement at a constant velocity is 100 N. With this information, I drew a force diagram for the tire. Since the car tire moves at a constant velocity, that means the forces are balanced, so the normal force is 130 N and the force of friction is -100 N. Using the equation Ff = Fn * µk, I solved for the coefficient of friction and found that µk = 0.769 for a car tire. Since the manufacturer says that the coefficient of friction for a truck tire is only 70% of that of car tires, µk = 0.538 for a truck tire.
Next, I drew force diagrams for both the car and the truck. I know the gravitational force because I know the weight of the car is 13600 N and the truck is 69700 N, so that is the magnitude of each of the gravitational forces. The normal force must balance out the gravitational force, and then I used the normal force and the two different coefficients of friction to find the force of friction acting on both the car and the truck. In order to find the mass of the car and truck, I just divided the weight by g, or 10. Then, using Newton's Second Law, I was able to find the acceleration of both the car and the truck after the collision, when the only horizontal force acting on them is the force of friction. After the collision, the car has an acceleration of -7.692 m/s^2 and the truck has an acceleration of -5.385 m/s^2.
Using these acceleration values, I used kinematic equations to find the velocity of the cars right after the collision. I know the acceleration, the distance traveled after the collision (8.2 m for the car and 11 m for the truck), and the final velocity (0 m/s because both come to a stop after their respective distances). Therefore, the third equation listed is the right one for me to use to solve for the initial velocity, or the velocity of the car and truck immediately after the crash. I found that the velocity for the car after the crash is 11.23 m/s, and the velocity for the truck after the crash is 10.88 m/s.
Now I need to break down these velocity values into the horizontal and vertical directions so I can find the horizontal and vertical momentum. I was given the angles of the direction the car and truck travel after they collide. This will allow me to break down the direction of the velocity. The equation for momentum is p = m * v. As you can see from my work, I used the angles 147° and 173° instead of 33° and 7° that way I could have the correct sign on my velocity values. Multiplying the velocity by the cosine of the angle for horizontal velocity and sine for vertical velocity, and then multiplying by the mass of the car and truck gives me the horizontal and vertical momentum for both the car and the truck. |
The conservation of linear momentum tells us that both the horizontal momentum and the vertical momentum of the system are conserved, and I can use this to find the initial momentum (right before the crash) of both cars. The truck was traveling in the x-direction and the car in the y-direction, so the truck originally had all of the horizontal momentum and the car had all of the vertical momentum. Therefore, I added the final horizontal and vertical momentums to find that the initial momentum of the truck was -88107.0 kg * m/s and the initial momentum of the car was 17564.7 kg * m/s.
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I can use the momentum of the truck to find the truck's velocity right before the crash
p = m * v
-88107 = 6970 * v
v = -12.64 m/s
The truck driver claimed that he was only traveling at 6.7 m/s, but he is actually traveling double this speed.
p = m * v
-88107 = 6970 * v
v = -12.64 m/s
The truck driver claimed that he was only traveling at 6.7 m/s, but he is actually traveling double this speed.
I can also find the velocity of the car right before the crash
p = m * v
17564.7 = 1360 * v
v = 12.92 m/s
I can use this velocity to test Mike Rokar's claim that he came to a full stop at the intersection. I know that the distance between the light and the crash was 13 m, and the maximum acceleration for the car is 3.0 m/s^2. Using kinematic equations, I can find that the velocity of the car at the traffic light was 9.423 m/s. This means that the car didn't come to a stop at the traffic light, so the car driver is lying as well, and both the car driver and the truck driver are at fault.
p = m * v
17564.7 = 1360 * v
v = 12.92 m/s
I can use this velocity to test Mike Rokar's claim that he came to a full stop at the intersection. I know that the distance between the light and the crash was 13 m, and the maximum acceleration for the car is 3.0 m/s^2. Using kinematic equations, I can find that the velocity of the car at the traffic light was 9.423 m/s. This means that the car didn't come to a stop at the traffic light, so the car driver is lying as well, and both the car driver and the truck driver are at fault.
Conclusion: Both the car driver and the truck driver lied in their claims; the car driver lied that he came to a complete stop at the traffic light, and the truck driver lied that he braked before the collision and was only traveling at 6.7 m/s before the crash. However, since the car driver had a flashing red light, he legally had to stop for the light, but he was traveling at a velocity of 9.423 m/s at the moment of the light. Therefore, he broke the law and lied more severely than the truck driver, so Mike Rokar the car driver is more at fault for the crash.