Concept explainers
February 3, 2009, was a very snowy day along Interstate 69 just outside of Indianapolis, Indiana. As a result of the slippery conditions and low visibility (50 yards or less), there was an enormous accident involving approximately 30 vehicles, including cars, tractor-trailers, and even a fire truck. Many witnesses said that people were driving too fast for the conditions and were too close together. In this problem, we explore two rules of thumb for driving in such conditions. The first is to drive at a speed that is half of what it would be in ideal conditions. The other is the “8-second” rule: Watch the vehicle in front of you as it passes some object such as a street sign, and you should pass that same object 8 seconds later. On a dry road, the 8-second rule is replaced by a 3-second rule. a. Assume vehicles on a slippery interstate highway follow both rules. What is the distance between the vehicles? b. If a driver followed the first rule of thumb, driving at a lower speed, but used the 3-second rule instead of the 8-second rule, what is the distance between the vehicles? How does that distance compare with the visibility on the day of the accident? c. Suppose drivers do not follow either rule of thumb for slippery conditions. What is the distance between vehicles? How does that distance compare with the visibility on that day? d. Suppose a driver was not obeying either rule of thumb when she sees a tractor-trailer that stopped on the highway. She presses on her brakes, locking the wheels, and her car crashes into the truck. Estimate the magnitude of the impulse exerted on her car. e. Estimate the impulse on the car in part (d) had the driver followed both rules of thumb for slippery conditions instead of ignoring them.
(a)
The distance between the vehicles when running through the slippery road.
Answer to Problem 78PQ
The distance between the vehicles when running through the slippery road is
Explanation of Solution
Assume that the speed of the vehicles in highway is
Write the expression for the distance travelled by the vehicle (the distance between the vehicles).
Here,
Conclusion:
Substitute
Therefore, the distance between the vehicles when running through the slippery road is
(b)
The distance between the vehicles according to the
Answer to Problem 78PQ
The distance between the vehicles according to the
Explanation of Solution
Given that the visibility on the day of accident is
Equation (I) gives the distance between the vehicles.
When the
Conclusion:
Substitute
The visibility on the day of accident in the units of meters is,
This indicates that the distance between the vehicles when the
Therefore, the distance between the vehicles according to the
(c)
The distance between the vehicles according to the
Answer to Problem 78PQ
The distance between the vehicles according to the
Explanation of Solution
Given that the visibility on the day of accident is
Equation (I) gives the distance between the vehicles.
In this case the vehicle is moving with speed of
Conclusion:
Substitute
This indicates that the distance between the vehicles when moving with speed of
Therefore, the distance between the vehicles according to the
(d)
The magnitude of impulse on the car.
Answer to Problem 78PQ
The magnitude of impulse on the car is
Explanation of Solution
According to the energy conservation, the kinetic energy of the vehicle at the crash equals the initial energy minus the energy dissipated by friction after applying the brakes.
Here,
Write the expression for the kinetic energy of the car at the moment of crash.
Here,
Write the expression for the initial kinetic energy of the car.
Here,
Write the expression for the change in thermal energy.
Here,
Use equation (III), (IV), and (V) in (II) and solve for
The impulse on the car is equal to the change in momentum. Since the truck stops the car after collision, the final momentum is zero. Thus, the magnitude of impulse on the car will be equal to the momentum of the car at the moment of collision.
Here,
The mass of the car be
Conclusion:
Substitute
Substitute
Therefore, the magnitude of impulse on the car is
(e)
The magnitude of impulse on the car with rules for the slippery road is applied.
Answer to Problem 78PQ
The magnitude of impulse on the car with rules for the slippery road is applied is
Explanation of Solution
When the rules for the slippery road is applied, the initial speed of the car would be
Equation (VI) gives the speed of the car at the moment of crash.
Equation (VII) gives the magnitude of impulse on the car.
Conclusion:
Substitute
Substitute
Therefore, the magnitude of impulse on the car with rules for the slippery road is applied is
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