Problem 53 - Thomas Calculus in SI Units

Question

Show that at some instant during a 2 -hour automobile trip the car's speedometer reading will equal the average speed for the trip.

Step-by-Step Solution

Answer

At some point during the trip, the car's speed will match the average speed due to the Intermediate Value Theorem.

Step 1: Understanding the Problem

We need to demonstrate that during a 2-hour trip, the car's speed (from the speedometer reading) will, at some point, equal the average speed for the entire trip. This is a specific application of the Intermediate Value Theorem.

Step 2: Define Average Speed

The average speed for the trip can be calculated by dividing the total distance traveled by the total time of the trip. Let's denote the total distance as \( D \) and the total time as 2 hours. Thus, the average speed \( v_{avg} = \frac{D}{2} \).

Step 3: Define Speed at a Given Instant

Define a function \( v(t) \), where \( v(t) \) is the speed of the car at time \( t \). This function describes the instantaneous speed at each point in time during the trip.

Step 4: Apply the Intermediate Value Theorem

Assume the speed function \( v(t) \) is continuous over the interval of the trip \([0, 2]\), meaning there are no sudden jumps in speed. According to the Intermediate Value Theorem, if \( v(0) eq v_{avg} \) and \( v(2) eq v_{avg} \), then there must exist some time \( c \) in the interval \((0, 2)\) where \( v(c) = v_{avg} \).

Step 5: Connect to Real World Scenario

In practical terms, since the speedometer reading varies continuously as the car accelerates and decelerates, it must, at some point during the 2-hour period, match the calculated average speed \( v_{avg} \).

Key Concepts

Average Speed

Average speed is a fundamental concept in motion, representing the total distance covered divided by the total time taken. If a car covers a distance of 120 miles in 2 hours, the average speed would be calculated as \( v_{avg} = \frac{120}{2} = 60 \text{ mph} \).

It's important to note that average speed does not account for variations in speed during the journey. A car might travel faster or slower at different times, but the average speed gives us a simple, overall picture of the journey's pace. By understanding average speed, you can get a basic measure of how much ground was covered relative to the time spent, like calculating how many miles are covered per hour during a trip.

Average speed is helpful when planning trips or when estimating the time it will take to travel a certain distance. Knowing this can help manage time better or optimize routes for efficiency.

Continuous Function

A continuous function is a mathematical concept where the output of the function changes smoothly and there are no abrupt jumps or breaks.

Think of a continuous line on a graph—it moves fluidly without any gaps or holes.

• Continuity ensures that the graph of the function is unbroken and you can draw it without lifting your pencil.
• This is crucial in real-world scenarios like speed readings, where you expect gradual changes rather than sudden shifts.

In the context of a car journey, we model the car's speed as a continuous function because a car doesn’t instantly go from one speed to another without passing through intermediary speeds.

The Intermediate Value Theorem relies on continuity to assert that if a function begins below and ends above a certain value, it must pass through that value in between.

Instantaneous Speed

Instantaneous speed is the speed of an object at a particular moment in time. Unlike average speed, which considers the total time and distance, instantaneous speed tells us how fast something is going "right now."

Imagine looking at a car's speedometer while driving. The number you see is the car's instantaneous speed at that very moment. It reflects the exact velocity of the vehicle at that point in time, capturing the real-time pace.

• Instantaneous speed can fluctuate as the car accelerates or decelerates.
• In calculus, it's often determined by evaluating the derivative of position with respect to time.

Knowing the instantaneous speed is essential for reacting to driving conditions and adjusting quickly to meet legal speed limits, enhancing driving safety.

Calculus

Calculus is a branch of mathematics that studies change—it helps us understand concepts like continuous functions and instantaneous speed.

Some key features of calculus relevant to the problem include:

• Derivatives, which help us find instantaneous rates of change, such as velocity or speed.
• The Intermediate Value Theorem, which helps in proving that within a continuous journey, a certain speed must be reached.

In the problem of driving a car, calculus allows us to make precise predictions about speed, direction, and overall movement by looking at the car's behavior over time. The Intermediate Value Theorem in calculus ensures that the car, traveling with a continuous speed function, will hit every speed between its starting and ending speeds at some point during the trip.

Thus, calculus not only helps in academic problems but also in understanding and solving real-world challenges by modeling and analyzing dynamic systems.