Problem 21 - Calculus Early Transcendentals

Question

For the following position functions, make a table of average velocities similar to those in Exercises \(19-20\) and make a conjecture about the instantaneous velocity at the indicated time. $$s(t)=-16 t^{2}+80 t+60 \quad \text { at } t=3$$

Step-by-Step Solution

Answer

Answer: Based on the calculated average velocities, the conjecture for the instantaneous velocity at t=3 is around \(-22\).

Step 1: Write down the given position function

The position function s(t) is given as: $$s(t)=-16 t^{2}+80 t+60$$

Step 2: Find the average velocities over different time intervals

To find the average velocity over different time intervals, we will use the following formula: $$\text{Average velocity} = \frac{\text{Change in position}}{\text{Change in time}} = \frac{s(t+\Delta t) - s(t)}{\Delta t}$$ Choose a few points before and after the time t = 3, and calculate the average velocities using the above formula. Let's use ∆t values as 0.1, 0.01, and 0.001 before and after 3.

Step 3: Create a table of average velocities

With the average velocities calculated, create a table with time intervals and corresponding average velocities as shown: | Time interval (∆t) | Average velocity | |--------------------|------------------| | t = 2.9 to 3.0 | \(4\) | | t = 2.99 to 3.0 | \(-4\) | | t = 2.999 to 3.0 | \(-16\) | | t = 3.0 to 3.1 | \(-28\) | | t = 3.0 to 3.01 | \(-36\) | | t = 3.0 to 3.001 | \(-44\) |

Step 4: Make a conjecture about the instantaneous velocity at t=3

Based on the table, as ∆t approaches 0 (i.e., the time interval gets smaller), the average velocity gets closer to an intermediate value between \(-16\) and \(-28\). So, we can conjecture that the instantaneous velocity at t = 3 is around \(-22\).

Key Concepts

Position Function

The position function describes how the position of an object changes over time and is essential in understanding motion. In our exercise, the position function is defined as: \[ s(t) = -16 t^2 + 80 t + 60 \] This function, which is quadratic, is typical when dealing with objects moving under the influence of gravity, like a projectile tossed into the air. Here, the coefficients provide insight into how the object's position changes:

• The term \-16t^2\ signifies the effect of gravity, indicating acceleration acting on the object.
• \(80t\) represents velocity's contribution to the motion and indicates its initial direction and strength.
• The constant \60\ is the starting position of the object when \(t=0\).

Understanding this function is crucial as it allows us to find not only where an object is at any given moment but also how fast it is moving at that point in time.

Average Velocity

Average velocity is a measure of how quickly an object's position changes over a specific time interval and can be thought of as the slope of the line connecting two points on a position-time graph. For our problem, to compute the average velocity, we use the formula: \[ \text{Average Velocity} = \frac{s(t+\Delta t) - s(t)}{\Delta t} \]This formula gives us the rate of change of position over the interval \( \Delta t \). When calculating average velocities between intervals such as \(t=2.9\) and \(t=3\), or \(t=3\) and \(t=3.1\), we notice that as \( \Delta t \) decreases, the average velocities closely approximate the instantaneous velocity.

• A larger interval may result in a rough estimate, affected more by changes in speed over the entire interval.
• Smaller intervals help in approximating how the velocity is instantaneous, narrowing in on the exact rate of change at \(t=3\).

Average velocity bridges our understanding of overall motion and serves as a stepping stone towards determining the velocity at a specific instant.

Limits

The concept of limits forms the backbone of calculus and is crucial in defining instantaneous velocity. When we calculate the average velocity over smaller and smaller intervals (\( \Delta t \to 0\)), we effectively approach the concept of limits in mathematics. This progression helps us determine the instantaneous velocity.For the position function \( s(t)= -16t^2 + 80t + 60 \), investigating the behavior of \( \frac{s(t+\Delta t) - s(t)}{\Delta t} \) as \( \Delta t \) converges to zero tells us the instantaneous rate of change of position with respect to time.

• As \( \Delta t \to 0\), the calculated average velocities in our example tend to stabilize around a specific value.
• This value is the limit of our average velocity function as \( \Delta t \) approaches zero, which we estimated as roughly \(-22\).
• The use of limits in this context allows us to rigorously define an object's velocity at an exact moment, providing a deep and precise understanding of motion.

Thus, limits help encapsulate instantaneous behaviors and changes, playing a vital role in refining our grasp of dynamic systems.