Delivering a Package by Air (Project Time Motion) help

In summary, the initial velocity of the package, v0, can be expressed as v0=v1+g(h/v0) where g is the magnitude of the acceleration due to gravity, h is the altitude of the airplane, and v1 is the initial velocity of the package relative to the airplane. This equation takes into account the relative velocities of the airplane and the package, as well as the effect of gravity on the package's trajectory.
  • #1
Kalie
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Delivering a Package by Air (Project Time Motion)...help

A relief airplane is delivering a food package to a group of people stranded on a very small island. The island is too small for the plane to land on and the only way to deliver the package is by dropping it. The airplane flies horizontally with constant speed at an altitude . The package is ejected horizontally in the negative x direction with speed relative to the plane. Assume is v1 less than v0. The positive x and y directions are defined in the figure.

|0000(plane)--->v0
|
|h
|
|__________0000(island)
D
Find the initial velocity of the package, v0, with respect to the ground.
Express the initial velocity of the package in terms of given quantities, v0,v1, h, and the magnitude of the acceleration due to gravity g, using x^ and y^ for the unit vectors in the x and y directions.

I figured the initial velocity of the package in the plane's frame of reference

vp=-v1x^

Velocity of the plane with respect to the ground

The frame of reference of the plane is moving with velocity +v0x^

I don't know what to do from there...
As you can tell I really don't get project time motion...
 
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  • #2
The "common" way of dealing with relative velocities is using subscripts:

[tex]v_{pg} = v_{pa} + v_{ag}[/tex]

this indicates that the (horizontal) velocity of the package relative to the ground is given by the sum of the velocity of the package relative to the airplane, v1, and the velocity of the airplane relative to the ground, v0.

Since the velocity of the package is in the opposite direction we get

[tex]v_{pg} = v0 - v1[/tex]

which will be a "positive" velocity in the sense that the package will have a "reduced" velocity in the direction of the motion of the airplane since it was thrown out towards the rear of the plane.
 
  • #3


I understand that project time motion involves analyzing the time and motion of a project or task. In this scenario, the project is delivering a package by air to a small island. The time and motion involved in this project can be calculated using principles of physics.

To solve this problem, we can use the equations of motion in two dimensions. The initial velocity of the package, v0, can be found by using the equation:

v0 = v1 - gt

Where v1 is the initial velocity of the package with respect to the plane, g is the acceleration due to gravity, and t is the time taken for the package to reach the ground. We can also use the equation of motion in the y-direction to find the time taken for the package to reach the ground:

h = v0yt - 1/2gt^2

Where h is the altitude of the plane, v0y is the initial velocity of the package in the y-direction, and t is the time taken for the package to reach the ground.

To find the initial velocity of the package with respect to the ground, we can use the Pythagorean theorem:

v0 = √(v0x^2 + v0y^2)

Where v0x and v0y are the initial velocities of the package in the x and y directions respectively.

In terms of given quantities, the initial velocity of the package can be expressed as:

v0 = √(v1^2 - 2ght + h^2)

Where v1, g, h, and t are known values.

In conclusion, understanding the principles of physics and using equations of motion can help us calculate the initial velocity of the package with respect to the ground in this project. By accurately calculating the time and motion involved, we can ensure a successful delivery of the package to the stranded individuals on the island.
 

1. How does delivering a package by air save time?

Delivering a package by air is significantly faster than other modes of transportation, such as ground or sea. This is because airplanes can travel at much higher speeds, allowing for quicker delivery times.

2. What factors affect the time it takes to deliver a package by air?

Some factors that may affect the time it takes to deliver a package by air include weather conditions, air traffic, and the distance between the origin and destination.

3. How can project time motion be used to improve the efficiency of delivering packages by air?

Project time motion can help identify any areas where the delivery process can be streamlined or improved. By analyzing the different steps involved in delivering a package by air, any bottlenecks or delays can be identified and addressed.

4. Is delivering a package by air more expensive than other methods?

In general, delivering a package by air is more expensive than ground or sea transportation. However, the cost may vary depending on the weight and size of the package, as well as the distance it needs to travel.

5. How can technology improve the delivery of packages by air?

New technology, such as drones and automated delivery systems, are being developed to improve the efficiency and speed of delivering packages by air. These advancements can also help reduce the cost of air delivery in the long run.

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