https://youtu.be/DI4WMqrYB4Q

In this video I go over another Applied Project example which my calculus book usually has at the end of the chapter. In this applied project, I go over modeling where a pilot should start the descent of a plane to land safely. This video illustrates in a simple way how real life plane landings are modeled.

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# Applied Project: Where should a Pilot Descent?

![Applied Project Pilot Descent.jpeg](https://files.peakd.com/file/peakd-hive/mes/ZqdLhQN5-Applied20Project20Pilot20Descent.jpeg)

## Where should a Pilot Descent?

An approach path for an aircraft landing is shown in the figure and satisfies the following
conditions:

(i) The cruising altitude is h when descent starts at a horizontal distance L from touchdown at the origin.

(ii) The pilot must maintain a constant horizontal speed v throughout descent.

(iii) The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity). 

![image.png](https://files.peakd.com/file/peakd-hive/mes/i5QKj1DC-image.png)

### Question 1

Find a cubic polynomial P(x) = ax<sup>3</sup> + bx<sup>2</sup> + cx + d that satisfies condition (i) by imposing suitable conditions on P(x) and P’(x) at the start of descent and at touchdown.

### Question 2

Use conditions (ii) and (iii) to show that

![image.png](https://files.peakd.com/file/peakd-hive/mes/oVfqJHxC-image.png)



### Question 3

Suppose that an airline decides not to allow vertical acceleration of a plane to exceed k = 860 mi/h<sup>2</sup> . 

If the cruising altitude of a plane is 35,000 ft and the speed is 300 mi/h, how far away from the airport should the pilot start descent?

### Question 4

Graph the approach path if the conditions stated in Problem 3 are satisfied.

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### Solution to Question 1

![image.png](https://files.peakd.com/file/peakd-hive/mes/TER3u9bX-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/SAweL2NT-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/vsCdWxKV-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/9hJxNueQ-image.png)

### Solution to Question 2

![image.png](https://files.peakd.com/file/peakd-hive/mes/I0uUoi1C-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/vLo5Jalz-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/si97w2X3-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/crJlu0k0-image.png)

### Solution to Question 3

![image.png](https://files.peakd.com/file/peakd-hive/mes/hHZam7I7-image.png)

### Solution to Question 4

![image.png](https://files.peakd.com/file/peakd-hive/mes/UaX6KUhp-image.png)