Problem 45 - Calculus Volume 1

Question

[T] 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: $$ \begin{array}{l}{\text { i. }[4,4.1]} \\ {\text { ii. }[4,4.01]} \\ {\text { iii. }[4,4.001]} \\ {\text { iv. }[4,4,0001]}\end{array} $$ 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 \(t=4\) is approximately 32 ft/s as the interval decreases.

Step 1: Understanding Average Velocity

The average velocity over a time interval \([a, b]\) is computed using the formula: \(\bar{v} = \frac{s(b) - s(a)}{b - a}\). Here, \(s(t)\) is the position function of the race car.

Step 2: Calculating Average Velocity for Interval [4, 4.1]

First, calculate \(s(4)\): \(s(4) = 8(4)^2 - \frac{1}{16}(4)^3 = 128 - 4 = 124\).Next, calculate \(s(4.1)\): \(s(4.1) = 8(4.1)^2 - \frac{1}{16}(4.1)^3 \approx 134.088 - 4.352 = 129.736\).Now, apply the average velocity formula: \(\bar{v} = \frac{129.736 - 124}{4.1 - 4} = \frac{5.736}{0.1} = 57.36\) ft/s.

Step 3: Calculating Average Velocity for Interval [4, 4.01]

Repeat the process for this interval:Calculate \(s(4.01)\): \(s(4.01) = 8(4.01)^2 - \frac{1}{16}(4.01)^3 \approx 128.3208 - 4.042 = 124.2788\).Apply the average velocity formula: \(\bar{v} = \frac{124.2788 - 124}{4.01 - 4} = \frac{0.2788}{0.01} = 27.88\) ft/s.

Step 4: Calculating Average Velocity for Interval [4, 4.001]

For this interval:Calculate \(s(4.001)\): \(s(4.001) = 8(4.001)^2 - \frac{1}{16}(4.001)^3 \approx 128.032008 - 4.008001 = 124.024007999\).Apply the average velocity formula: \(\bar{v} = \frac{124.024007999 - 124}{4.001 - 4} = \frac{0.024007999}{0.001} = 24.008\) ft/s.

Step 5: Calculating Average Velocity for Interval [4, 4.0001]

Last interval:Calculate \(s(4.0001)\): \(s(4.0001) = 8(4.0001)^2 - \frac{1}{16}(4.0001)^3 \approx 128.00320008 - 4.00008 = 124.00312008\).Apply the average velocity formula: \(\bar{v} = \frac{124.00312008 - 124}{4.0001 - 4} = \frac{0.00312008}{0.0001} = 31.2008\) ft/s.

Step 6: Determining Instantaneous Velocity at t=4

As the length of the time interval approaches zero, the average velocity approaches the instantaneous velocity of the car at \(t=4\) seconds. From the calculated average velocities, the instantaneous velocity is evident in the limiting process. It appears to approach approximately 32 ft/s, extrapolating from how the values change as the intervals get smaller.

Key Concepts

Average Velocity

The concept of average velocity is crucial to understanding movement over a specific period. To compute the average velocity, you must determine how much the position of an object has changed over a set period of time, also known as a time interval.

It is calculated using the formula:

• \( \bar{v} = \frac{s(b) - s(a)}{b - a} \)

Here, \(s(a)\) and \(s(b)\) represent the position of the object at times \(a\) and \(b\), respectively.

Average velocity helps illustrate how quickly an object is moving on average over a specified time interval. It's essential when the speed of an object isn't constant, allowing us to average out its speed over time.

Position Function

The position function of an object describes its location in relation to time. It is a powerful tool in physics and calculus.

For example, the position function for the race car in the exercise is:

• \( s(t) = 8t^{2} - \frac{1}{16}t^{3} \)

This function plugs in any time \(t\) and outputs the position of the car at that exact moment.

Understanding the position function allows you to calculate how far an object has traveled from a certain point at any given time by simply substituting the time variable \(t\) into the equation.

Time Interval

In the context of physics and missions involving motion, a time interval refers to the difference between two points in time. It often determines the bounding of observations for measuring changes or developing predictions about an object's motion.

For example, in calculating average velocity, different time intervals such as

• \([4, 4.1]\)
• \([4, 4.01]\)
• \([4, 4.001]\)
• \([4, 4.0001]\)

were used to see the behavior of velocity as the interval gets smaller. Understanding time intervals is important for analyzing situations where the object does not move at a constant speed.

Limiting Process

The limiting process involves examining what happens as a variable approaches a certain value. It's fundamental in calculus for understanding instantaneous phenomena.

In this exercise, we looked at how the average velocity changes as the time interval shrinks towards zero. Essentially, as we calculated the average velocity over smaller and smaller intervals, we aimed to approach the instantaneous velocity, which occurs at a single, exact point in time.

Through the limiting process:

• The average velocity gives us a more accurate depiction of instantaneous velocity as the interval narrows.
• The calculated values from different intervals start converging to a particular number when the interval approaches zero.

This is how we deduced that the instantaneous velocity at \(t=4\) seconds is around 32 ft/s, showing the power of limits in understanding real-world motion.