Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

A car is stationary at a toll booth. Twenty minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Explain why the car must have exceeded 60 miles per hour at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

Step-by-Step Solution

Answer

The car exceeded 60 mph to maintain an average speed of 60 mph from rest.

Step 1: Understand the Problem

We have a car that starts from rest (0 mph) and reaches a speed of 60 mph at a point 20 miles away within 20 minutes. We need to determine whether the car must have exceeded 60 mph during this trip.

Step 2: Calculate the Average Speed

To calculate the average speed of the car, we use the formula for average speed: \[\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}.\]The total distance is 20 miles, and the total time is 20 minutes, which is \(\frac{1}{3}\) of an hour. Thus, the average speed is \(\frac{20}{\frac{1}{3}} = 60\) miles per hour.

Step 3: Apply the Mean Value Theorem

The Mean Value Theorem (for velocities) states that if a vehicle travels a certain distance in a given time, at some point during the journey, the instantaneous speed must equal the average speed. Since the car starts from rest and is later traveling at 60 mph on average, its acceleration at some point must have exceeded 60 mph to reach an average of 60 mph.

Step 4: Analyze the Speed Dynamics

Initially, the car is at 0 mph, then later recorded at 60 mph with an average speed of 60 mph over the journey. However, to achieve this average given that it started from rest, at some point it must have been traveling faster than 60 mph to compensate for the earlier slower speeds and still maintain the average.

Step 5: Conclude with a Logical Argument

Because the car reached 60 mph at the end while having an average speed of 60 mph over the entire journey, it must have exceeded 60 mph at some point to balance out the other instances when it was traveling slower than 60 mph initially.

Key Concepts

Average Speed

Average speed is a fundamental concept in physics and everyday life. It helps us understand how fast something is moving on average, over a certain distance or period of time. It is essential for comparing speeds over journeys or trips.
This is calculated using the formula:

• \( \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \)

For instance, in a trip covering 20 miles in 20 minutes, one can compute the average speed by converting minutes into hours, as speed is commonly expressed in miles per hour (mph).
Therefore, 20 minutes is approximately \( \frac{1}{3} \) of an hour. The car's average speed during this period is \( \frac{20}{\frac{1}{3}} = 60 \) mph.

Instantaneous Speed

While average speed gives a broad view over a journey, instantaneous speed tells us how fast an object is traveling at any specific moment.
Unlike average speed, which is calculated over a distance or time interval, instantaneous speed is precise. It is similar to what a car's speedometer displays.
In the given scenario where a car is recorded at 60 mph at a particular point, the instantaneous speed at that moment is 60 mph.
This speed can vary throughout the trip due to accelerations and decelerations. At times, the instantaneous speed would be greater than the average speed to compensate for periods at slower speeds.

Velocity

Velocity is often confused with speed, yet it's a slightly more complex concept.
While speed is how fast something moves, velocity includes direction.
It is a vector quantity, which means it has both magnitude and direction.

• For example, heading north at 60 mph is a velocity, not just speed.

In the car journey, if we specify a direction (like north along a highway), we can then say the car has a velocity of 60 mph north when it was clocked at that speed.
Therefore, both speed and direction should be considered when discussing velocity in trip scenarios.

Acceleration

Acceleration describes how quickly the speed of an object is changing.
When a car accelerates, it means it is increasing its speed over time.
Acceleration can be positive (speeding up) or negative (slowing down).

• In our example, the car starts from rest at the toll booth. It must increase its speed to reach 60 mph and continue that until reaching the 20-mile point.

Because the average speed was consistent with the speed recorded at one point, the car likely accelerated past 60 mph between starting and ending points to balance any periods of slower travel needed to start from zero. This ensures a consistent average speed. This showcases how important acceleration is in tying together instantaneous speeds at multiple points to the overall average.