2. Falling object with Atmospheric Drag: Assume a zero initial velocity and that after 2 seconds the velocity is 19 m/s . For a 10 kg object, with a gravitational acceleration of 10 (m/s)/s we have the falling object with drag equation: dv = 10 dt k 10 a. Use a degree 5 Taylor Series centered about zero and the fact that the velocity is 19 m/s after 2 seconds to get a quadratic equation in terms of k, then use the smaller solution to approximates k, to three decimal places. b. With vour estimate of k from above in place use separation of variables to find v(t)
2. Falling object with Atmospheric Drag: Assume a zero initial velocity and that after 2 seconds the velocity is 19 m/s . For a 10 kg object, with a gravitational acceleration of 10 (m/s)/s we have the falling object with drag equation: dv = 10 dt k 10 a. Use a degree 5 Taylor Series centered about zero and the fact that the velocity is 19 m/s after 2 seconds to get a quadratic equation in terms of k, then use the smaller solution to approximates k, to three decimal places. b. With vour estimate of k from above in place use separation of variables to find v(t)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Falling Object with Atmospheric Drag
Consider the following scenario for a falling object in the presence of atmospheric drag:
Assume a **zero initial velocity** and that **after 2 seconds the velocity is 19 m/s**.
For a 10 kg object, with a gravitational acceleration of 10 (m/s²), the equation for the falling object with drag is given by:
\[
\frac{dv}{dt} = 10 - \frac{k}{10}v^2
\]
#### a. Taylor Series Approximation:
Use a degree 5 Taylor Series centered about zero and the fact that the velocity is 19 m/s after 2 seconds to derive a quadratic equation in terms of \( k \). Then, use the smaller solution to approximate \( k \) to three decimal places.
#### b. Separation of Variables:
With the estimated value of \( k \) from part (a), use the method of separation of variables to find \( v(t) \). Given \( v(0) = 0 \), determine the solution and then compare how well your model estimates \( v(2) \) with the calculated \( k \).
This method helps to understand the dynamics of a falling object considering both gravitational and resistive (drag) forces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22499db0-d22f-4756-b954-120799dc3fe6%2F1b0afdde-aa1b-4aba-94d9-125325239242%2Fcgxemz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Falling Object with Atmospheric Drag
Consider the following scenario for a falling object in the presence of atmospheric drag:
Assume a **zero initial velocity** and that **after 2 seconds the velocity is 19 m/s**.
For a 10 kg object, with a gravitational acceleration of 10 (m/s²), the equation for the falling object with drag is given by:
\[
\frac{dv}{dt} = 10 - \frac{k}{10}v^2
\]
#### a. Taylor Series Approximation:
Use a degree 5 Taylor Series centered about zero and the fact that the velocity is 19 m/s after 2 seconds to derive a quadratic equation in terms of \( k \). Then, use the smaller solution to approximate \( k \) to three decimal places.
#### b. Separation of Variables:
With the estimated value of \( k \) from part (a), use the method of separation of variables to find \( v(t) \). Given \( v(0) = 0 \), determine the solution and then compare how well your model estimates \( v(2) \) with the calculated \( k \).
This method helps to understand the dynamics of a falling object considering both gravitational and resistive (drag) forces.
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