Question

Compare carefully to Exercise \(33 .\) A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another officer \(30 \mathrm{mi}\) along the highway. When the car reached the location of the second officer 30 min later, it was clocked going 60 mi/hr. Can the patrol officer conclude that the driver exceeded the speed limit?

Short answer

Answer: No, we cannot definitively conclude if the driver exceeded the speed limit, as the speed limit was not provided in the given information.

Step-by-step solution

Calculate the average speed of the car

We can calculate the average speed by dividing the total distance traveled by the time taken. The car traveled 30 miles in 30 minutes (0.5 hours). The formula for average speed is: Average speed = \(\frac{\text{total distance}}{\text{total time}}\)

Plug in the values and calculate the average speed

Now, let's plug in the values: Average speed = \(\frac{30 \text{ miles}}{0.5 \text{ hours}}\) Average speed = 60 mi/hr

Compare the average speed to the speed limit

In this problem, we don't have the information about the speed limit, so we cannot conclude whether the driver exceeded the speed limit or not based on the given information. However, we can say that if the speed limit was above 60 mi/hr, then the driver did not exceed the speed limit, but if the speed limit was below 60 mi/hr, then the driver exceeded the speed limit.

Key concepts

Average Speed Calculation

Understanding how to calculate the average speed is essential when examining motion, as it provides information on how fast an object has been moving over a period of time. The concept is elegantly simple: you take the total distance traveled and divide it by the total time it took to travel that distance. This can be denoted by the formula:

\[\begin{equation}\text{Average speed} = \frac{\text{total distance}}{\text{total time}}\end{equation}\]
In the exercise, we apply this formula to understand a car's movement. The car travels a distance of 30 miles in a time of 0.5 hours (30 minutes). Thus, the average speed calculation becomes a straightforward arithmetic problem:

\[\begin{equation}\text{Average speed} = \frac{30 \text{ miles}}{0.5 \text{ hours}}\end{equation}\]
which simplifies to 60 miles per hour (mph). It's important to note that average speed does not tell us about the variations in speed during the trip – it only gives us a single number representing the overall rate of motion.

Calculus Applications

Calculus has numerous applications in understanding the dynamics of motion. In the context of speed and velocity, calculus allows us to analyze changes over time in a more nuanced way than simply calculating average speed.

For instance, if we wanted to know whether the car in the exercise accelerated or decelerated, we would use differentiation, a fundamental tool in calculus, to find the rate of change of the car's velocity. Conversely, if we knew the car's acceleration at various points in time, we could use integration to find the car's velocity and position over time.

Instantaneous Velocity vs. Average Speed

Calculus provides the tools to distinguish between the average speed of the car and its instantaneous velocity – the speed at a specific point in time. The average speed over an interval may be the same even if a car starts at rest, accelerates, and then decelerates. Calculus allows us to delve into the car's speed profile for the entire journey, providing a detailed understanding of the car's behavior on the road.

Interpreting Speed

Interpreting speed involves more than just understanding its numerical value; it's about context and implications. In the case of the patrol officer and the speeding car, interpreting the average speed means considering the possible variations in speed during the 30-minute period.

While we calculated the average speed to be 60 mi/hr, we don't necessarily know the driver's speed at every moment of the trip. For example, the driver could have been driving below the speed limit for most of the journey, then sped up to 60 mi/hr just before being clocked by the second officer.

Speed Limits and Safety

Interpreting speed also requires taking safety and legal limits into account. If the speed limit is 60 mi/hr, the average speed being equal to the speed limit does not guarantee that the speed limit was not exceeded at some point. Roads are designed with a maximum safe operating speed in mind, and exceeding this can increase the risk of accidents. Therefore, in the absence of specific data points during the trip, conclusions about following speed regulations based solely on average speed need to be made cautiously.