Math and Science Final Report

Purpose

The final solution must be supported by many reasons for its selection. The past weeks have been dedicated to refining my developmental work for the final solutions. These steps are necessary for the production of the final solution. Mistakes need to be avoided as much as possible. The math, science, and technology related to the final solution will define the project and support the choice of solutions.

With respect to my particular project, I will need to perform more in depth research for the content of the animation – the theory of relativity. Also, I will need to research in more detail the timing schemes of the animation.

Science: The Theory of Relativity

Motion is Relative

Whenever we discuss motion, we must specify the position from which motion is being observed and measured. For example, if someone is walking 1 km/h on an airplane, they are only moving at this speed relative to the airplane when they are moving at much greater speed to outside observers. Technically, an outside observer that sees you walking in the cabin of an airplane sees you moving faster than the airplane itself. Speed is a relative value and depends on the place (or frame of reference) from which the speed is being recorded.

So, suppose your friend can always pitch a ball at the same speed of 60 km/h. Now, suppose your friend pitches the ball at this speed while moving towards you at 40 km/h on the bed of truck. When the ball meets you, it will be moving at 100 km/h. Now suppose the truck moves away from you at the same speed of 40 km/h as your friend pitches the ball. Subtracting the two speeds, you will find that you will now receive the ball at only 20 km/h (Figure 1).

Figure 1 – the relativity of a baseball as applied to a baseball thrown in the same direction as a moving vehicle from that vehicle.

This concept is not exclusive the realm of modern physics – physicists as early as Galileo noticed this trend in motion; however, it wasn’t until the time of Einstein when he expanded the relativity of speed in the context of spacetime.

The Speed of Light is Constant

In the previous discussion, we have established that physical objects – whether they be baseballs or people – only have speeds that are calculated relative to an observer. However, light seems to defy this property. When an observer is in any frame of reference, the speed of light measured in a different or identical frame of reference relative to the observer always will have the same value, often called the variable c (Figure 2). The speed of light has been approximated to equal 3 x 10^8 m/s.

Figure 2 – When the speed of light is measured in any frame of reference, the same value, c, will always be obtained.

This property of light was found by two American physicists in 1887. A.A. Michelson and E.W. Moreley performed an experiment to determine the differences in the speed of light in different directions. They though the emotion of the earth in its orbit around the sun would cause shifts in the speed of light. The speed should have been faster when light traveled in the same direction of the earth, and slower when it traveled at right angles to the earth. Using a device called an interferometer, they found that thee speed was the same in all directions.

The First Postulate of Special Relativity

“All laws of nature are the same in uniformly moving frames of reference.”

There is no stationary hitching post in the universe to which speed can be calculated against – it can only be found relative to an observer. So, when a space ship shoots away and says that the earth is flying away from him, he is just as correct as the earthlings who say that the space ship is moving away from the earth.

Also, on a uniformly (constant speed) moving frame of reference, the same laws of physics apply to different moving frames of reference. For example, a ball dropped in a constantly moving car or an airplane behaves the same as one that is dropped from a stationary person on the surface of the earth. Any number of experiments can be devised to detect accelerated motion, but none can be devised to detect a state of uniform motion.

The Second Postulate of Special Relativity

“The speed of light in empty space will have the same value regardless of the motion of the source or the observer.”

This property of light has been discussed in the previous section. It is this property of light that unifies space and time.

Time Dilation

Einstein offered a solution for the seemingly contradictory properties of the constant speed of light and the relativity of speed of everything else. He through out the idea that time was an absolute, and that the rate at which time passes is also actually relative. But how can this be?

Imagine an empty tube with a mirror at each end. A mirror of light bounces back and forth between the parallel mirrors (Figure 3). The mirrors are perfect reflectors, so the flash bounces indefinitely. So, if the tube is 300 000 km in length, then each bounce will take 1 s in the frame of reference of the light clock. If the tube is 3 km long, each bounce will take .00001 seconds.

Figure 3 – The “light clock”

Suppose we view this light clock as it whizzes by while being observed from a relatively stationary frame of reference. This observer will notice that the light flash will bounce up and down along a longer diagonal path than the path that the light clock perceives (Figure 4).

Figure 4 – An observer who is passed by the moving ship observes the flash moving along a diagonal path.

Since the speed of light will not increase, we must measure more time between bounces over longer perceived distances. The stationary observer can only deduce that the light clock on the moving space ship moves slower through time than a light clock that is within his own frame of reference. The slowing of time is not peculiar to the light clock – it is time itself in the moving frame of reference, and everything that occurs and exists within that reference moves slower through time as well. A faster moving body does not perceive “slow motion” or live a longer life. According to Einstein’s first postulate, al laws of nature are the same in all uniformly moving frames of reference. A rapidly moving person will have no clue that it seems his time is dilated when viewed from other frames of reference.

Mathematics: Time Dilation and Frame Computation

Time Dilation Equation

, where t₀ is the time measured by the observer of an alternative frame of reference where time is measured as t.

Frame Computation

The slide show will be created at 30 frames per second.

Following the principle of Frame rate x Time = Frame number, in order to properly calculate the frame number at a position of time t, they must use this equation:

Technology: Adobe Flash Software

• Pros
• Can create simple, vector based diagrams
• Very small file sizes
• Can be encoded into video format through software
• Cartoon style creates a more vibrant presentation
• Software can “smooth out” drawings
• “Tweening” makes animation simpler
• Allows for integration with Adobe Suite of products
• Can easily export in a number of movie and image formats
• Fast rendering times
• Cons
• Cartoon style look may turn off some viewers
• No realistic look to drawings
• For the most part, limited to two dimensional drawings
• Quality of video conversion may be sub par
• Coding in the software can become quite technical and tedious
• Requires user to draw most objects – imported images do not behave as well as objects drawn within the software
• May require a tablet to create best drawings (instead of mouse)

Figure 5 – A screenshot of the Adobe Flash user interface

Adobe Flash uses "key frames "in order to animate. Objects are set at certain positions at certain frames; motion paths are automatically created to get from point A in frame A to point B in frame B. The image to the left shows the drawing interface, and at the top is the time line that shows frame positions. Flash documents can be easily saved as shockwave files (.swf, a regular flash file for a web browser), animated gifs, movie files, or individual images. None of these exporting methods requires third party software. To the left is an example of an exported animation of a ship, similar to the one created in PowerPoint in the previous brainstorming section. Lines can be drawn with a mouse, or more popularly, with a third-party tablet attached the computer via USB interface. Also, there are tools to create ovals, rectangles, and straight lines. Objects can be separated to different layers; these layers can be locked or made invisible to make an easier drawing environment to work with. Also, each layer has its own time line for frames, which makes it easy to animate only within a certain layer while keeping a background layer still. Ironically, Adobe Flash is a complicated program used to make more simple animation.

Conclusion

These scientific concepts discussed will be used within the slide show itself. The mathematical equation for time dilation may be incorporated into the animation as well, but only if the time allows for it. The mathematical formula for finding frame position will be used during the production of the animation in order to see what time point I am working on in an animation and how fast I should be creating changes in motion between each frame. And lastly, the software being used, Adobe Flash, will be the principle tool used to create the animation – mastery of this software is key to the success of the final solution.