### Falling Object with Atmospheric DragConsider 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.

initial velocity= 0 m/s velocity after 2 seconds= 19 m/s m= 10 kg g(gravitational acceleration)= 10…

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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Question

### 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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