Problem 45 - Calculus Volume 1

Question

The position in feet of a race car along a straight track after \(t\) seconds is modeled by the function \(s(t)=8 t^{2}-\frac{1}{16} t^{3}\). a. Find the average velocity of the vehicle over the following time intervals to four decimal places: i. [4,4.1] ii. [4,4.01] iii. [4,4.001] iv. [4,4.0001] b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at \(t=4\) seconds.

Step-by-Step Solution

Answer

The instantaneous velocity at t=4 seconds is 88 feet/second.

Step 1: Understand Average Velocity

The average velocity of a vehicle over a given time interval [a, b] is calculated using the formula: \( v_{avg} = \frac{s(b) - s(a)}{b-a} \) where \( s(t) \) is the position function.

Step 2: Calculate Position at Each Time Point

For each time interval, you need to calculate the position of the vehicle at both ends of the interval. Use the position function \( s(t) = 8t^2 - \frac{1}{16}t^3 \) to find these values. Let's first calculate at each endpoint of our intervals.

Step 3: Calculate Position at t=4

Plug \( t = 4 \) into the position function: \[ s(4) = 8(4)^2 - \frac{1}{16}(4)^3 = 128 - \frac{1}{16} \times 64 = 128 - 4 = 124 \text{ feet} \]

Step 4: Calculate Positions for Other Interval Ends

Find the position for \( t = 4.1, 4.01, 4.001, \text{ and } 4.0001 \) respectively.- \( s(4.1) = 8(4.1)^2 - \frac{1}{16}(4.1)^3 \approx 132.3026 \text{ feet} \)- \( s(4.01) = 8(4.01)^2 - \frac{1}{16}(4.01)^3 \approx 124.8801 \text{ feet} \)- \( s(4.001) = 8(4.001)^2 - \frac{1}{16}(4.001)^3 \approx 124.0880 \text{ feet} \)- \( s(4.0001) = 8(4.0001)^2 - \frac{1}{16}(4.0001)^3 \approx 124.0088 \text{ feet} \)

Step 5: Calculate Average Velocities

Now, compute the average velocity over each interval using the position values:- For [4, 4.1]:\[ v_{avg} = \frac{132.3026 - 124}{4.1 - 4} = \frac{8.3026}{0.1} = 83.026 \text{ feet/sec} \]- For [4, 4.01]:\[ v_{avg} = \frac{124.8801 - 124}{4.01 - 4} = \frac{0.8801}{0.01} = 88.01 \text{ feet/sec} \]- For [4, 4.001]:\[ v_{avg} = \frac{124.0880 - 124}{4.001 - 4} = \frac{0.0880}{0.001} = 88 \text{ feet/sec} \]- For [4, 4.0001]:\[ v_{avg} = \frac{124.0088 - 124}{4.0001 - 4} = \frac{0.0088}{0.0001} = 88 \text{ feet/sec} \]

Step 6: Conclude Instantaneous Velocity

The average velocities approach 88 feet/second as the interval shrinks towards \( t = 4 \). Hence, the instantaneous velocity of the vehicle at \( t = 4 \) seconds is 88 feet/second.

Key Concepts

Instantaneous Velocity

Instantaneous velocity refers to the speed and direction of an object at a specific instant in time. Imagine driving a car and looking at the speedometer, which shows how fast you are going at that very moment. This is what we mean by instantaneous velocity.
It's different from average velocity, which accounts for the total distance traveled over a wider timespan. In physics and calculus, the concept of instantaneous velocity is crucial because it allows us to understand motion at very precise points in time.

• To find instantaneous velocity using calculus, you generally need to take the derivative of the position function, which provides the rate of change of position with respect to time.
• This derivative, denoted as \( s'(t) \), represents the instantaneous velocity at time \( t \).
• In this case, the exercise concludes that as the average velocity approaches a certain value on shrinking time intervals, the instantaneous velocity at \( t = 4 \) seconds is 88 feet/second.

Position Function

The position function, typically denoted as \( s(t) \), describes the position of an object at any given time \( t \). This mathematical model reveals how an object's location on a line or in space changes over time.
In this scenario, the position function given is \( s(t) = 8t^2 - \frac{1}{16}t^3 \), which provides the position of the race car along the track for different times. By substituting specific times into this function, one can calculate exact positions.

• For example, when \( t = 4 \), plug it into the function: \( s(4) = 8 \times 4^2 - \frac{1}{16} \times 4^3 = 124 \text{ feet} \).
• This calculation shows the precise position of the car at 4 seconds, necessary for computing average velocities over various intervals.

The position function is a fundamental concept used to model real-world motion, and helps in determining velocities.

Time Intervals

Time intervals are periods between two points in time within which changes can be measured. In the context of this problem, they help calculate average velocities by providing the start and end times.

• The formula \( v_{avg} = \frac{s(b) - s(a)}{b-a} \) is employed to compute the average velocity, where \([a, b]\) indicates the interval.
• For instance, the interval \[4, 4.1\], results in an average velocity of 83.026 feet/second.
• Narrowing the intervals gradually towards a single point, \( t = 4 \), reveals vital insights about instantaneous speeds.

Smaller intervals give a better approximation of the instantaneous velocity, making time intervals crucial in determining velocity trends.

Calculus

Calculus is a branch of mathematics focused on change and motion, often utilizing derivatives and integrals. It enables us to handle problems involving continuously changing quantities by providing tools to calculate specific rates of change (like velocity). In understanding velocity:

• The derivative of the position function, \( s'(t) \), represents the instantaneous velocity, illustrating calculus' ability to express motion dynamics precisely.
• In the given problem, the convergence of average velocities across decreasing intervals provides a calculus-based indication of instantaneous velocity. As the average velocities reach 88 feet/second, they closely approximate the derivative at \( t = 4 \).

Thus, calculus beautifully bridges our understanding of static mathematical equations with dynamic real-world phenomena like the race car's motion, making it indispensable for comprehending physics and engineering concepts.