In an auto accident, a car hit a pedestrian and the driver then slammed on the brakes to stop the car. During the subsequent trial, the driver's lawyer claimed that the driver was obeying the posted 35.0 mph speed limit, but that the limit was too high to enable him to see and react to the pedestrian in time. You have been called as the state's expert witness. In your investigation of the accident site, you make the following measurements: The skid marks made while the brakes were applied were 325 ft long, and the tread on the tires produced a coefficient of kinetic friction of 0.300 with the road.
If the driver's speeding ticket is $10 for each mile per hour he was driving above the posted speed limit, would he have to pay a ticket, and if so, how much would it be?
To determine whether the driver was speeding, we first need to calculate the car's initial speed before the brakes were applied. We can use the equation:
vf^2 = vi^2 + 2ad
where:
- vf is the final velocity (0 mph since the car stops)
- vi is the initial velocity (what we need to find)
- a is the deceleration (which can be calculated using the coefficient of kinetic friction)
- d is the distance traveled during deceleration (the length of the skid marks)
Rearranging the equation, we get:
vi = sqrt(2ad)
Given:
- a = coefficient of kinetic friction = 0.300
- d = length of the skid marks = 325 ft
Plugging in the values, we can calculate the initial speed (vi):
vi = sqrt(2 * 0.300 * 32.2 ft/s^2 * 325 ft)
vi ≈ sqrt(1959)
vi ≈ 44.26 ft/s ≈ 30.18 mph
The driver's initial speed before applying brakes was around 30.18 mph.
Now we can compare this speed with the posted speed limit of 35.0 mph. Since the driver was driving below the speed limit, there would be no speeding ticket fines to pay.
To determine if the driver would need to pay a ticket and the amount, we need to calculate the driver's actual speed at the time of the accident, compare it to the posted speed limit, and calculate the difference.
First, let's determine the driver's initial speed before applying the brakes. We can use the equation of motion:
v^2 = u^2 + 2as
Where:
- v is the final velocity (0 mph since the car comes to a stop).
- u is the initial velocity (unknown).
- a is the acceleration due to braking (which we'll calculate).
- s is the distance covered during braking (325 ft or 0.0621 miles, given).
Rearranging the equation:
u^2 = v^2 - 2as
Substituting the given values:
u^2 = 0 - 2 * (0.300) * 0.0621
u^2 = -0.03726
Since velocity cannot be negative, it indicates that the car was already immobilized when the brakes were applied. Therefore, we can infer that the car was already stationary or moving at an extremely low speed. This suggests that the driver wasn't speeding, and there would be no basis for a speeding ticket.
Hence, the driver would not have to pay a ticket for speeding.
However, it should be noted that this analysis assumes that the braking force was solely due to the coefficient of friction. Other factors, such as the condition of the brakes and the weight of the vehicle, may also have been at play in reality.