Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Let's consider the position function [tex]\( s(t) = 8 \sin(3t) \)[/tex], which describes a block bouncing vertically on a spring.
To find the average velocity over a given time interval [tex]\([t_0, t_1]\)[/tex], we use the formula:
[tex]\[ \text{Average Velocity} = \frac{s(t_1) - s(t_0)}{t_1 - t_0} \][/tex]
We need to calculate the average velocities over various time intervals and fill in the table. Let's list the intervals and then their corresponding average velocities.
The time intervals are:
1. [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex]
2. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1\right]\)[/tex]
3. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01\right]\)[/tex]
4. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001\right]\)[/tex]
5. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001\right]\)[/tex]
Using the given position function [tex]\( s(t) \)[/tex], let's calculate the average velocities for each interval:
- For the interval [tex]\( \left[\frac{\pi}{2}, \pi\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 5.092958178940653 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 3.5730808699515273 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 0.3599730008099652 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 0.03599997299997112 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 0.0035999999692397167 \][/tex]
We can now fill in the table with these results.
[tex]\[ \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Time \\ Interval \end{tabular} & \begin{tabular}{c} Average \\ Velocity \end{tabular} \\ \hline \(\left[\frac{\pi}{2}, \pi\right]\) & 5.092958178940653 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1\right]\) & 3.5730808699515273 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01\right]\) & 0.3599730008099652 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001\right]\) & 0.03599997299997112 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001\right]\) & 0.0035999999692397167 \\ \hline \end{tabular} \][/tex]
From the table, we observe that as the interval gets smaller and smaller, the average velocity approaches zero. Hence, we can make the conjecture that the instantaneous velocity at [tex]\( t = \frac{\pi}{2} \)[/tex] is approximately zero. This result is confirmed by the smallest average velocities in the list above, suggesting the instantaneous velocity value is [tex]\(-4.408728476930472 \times 10^{-15}\)[/tex], which is extremely close to zero.
To find the average velocity over a given time interval [tex]\([t_0, t_1]\)[/tex], we use the formula:
[tex]\[ \text{Average Velocity} = \frac{s(t_1) - s(t_0)}{t_1 - t_0} \][/tex]
We need to calculate the average velocities over various time intervals and fill in the table. Let's list the intervals and then their corresponding average velocities.
The time intervals are:
1. [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex]
2. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1\right]\)[/tex]
3. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01\right]\)[/tex]
4. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001\right]\)[/tex]
5. [tex]\(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001\right]\)[/tex]
Using the given position function [tex]\( s(t) \)[/tex], let's calculate the average velocities for each interval:
- For the interval [tex]\( \left[\frac{\pi}{2}, \pi\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 5.092958178940653 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 3.5730808699515273 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 0.3599730008099652 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 0.03599997299997112 \][/tex]
- For the interval [tex]\( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001\right] \)[/tex]:
[tex]\[ \text{Average Velocity} = 0.0035999999692397167 \][/tex]
We can now fill in the table with these results.
[tex]\[ \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Time \\ Interval \end{tabular} & \begin{tabular}{c} Average \\ Velocity \end{tabular} \\ \hline \(\left[\frac{\pi}{2}, \pi\right]\) & 5.092958178940653 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1\right]\) & 3.5730808699515273 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01\right]\) & 0.3599730008099652 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001\right]\) & 0.03599997299997112 \\ \hline \(\left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001\right]\) & 0.0035999999692397167 \\ \hline \end{tabular} \][/tex]
From the table, we observe that as the interval gets smaller and smaller, the average velocity approaches zero. Hence, we can make the conjecture that the instantaneous velocity at [tex]\( t = \frac{\pi}{2} \)[/tex] is approximately zero. This result is confirmed by the smallest average velocities in the list above, suggesting the instantaneous velocity value is [tex]\(-4.408728476930472 \times 10^{-15}\)[/tex], which is extremely close to zero.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
