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If you ve used graphics software such as Photoshop or Gimp, you re probably familiar with some of these blend modes. Each of them uses a particular formula that takes the red, green, and blue components of each of the two colors and produces an output color. We won t describe the blend modes here, but you can find detailed descriptions and examples of them at http://en.wikipedia.org/wiki/Blend_modes and elsewhere. What we will do here is show you how to build an application that gives the user a hands-on way to explore all of these modes, two chosen colors at a time. The blend modes we ll be using are all built into Core Graphics, an important part of Cocoa's graphics architecture that we ll talk more about in 13. Using this builtin functionality lets you skip doing any sort of color computation on your own; all you do is call Core Graphics functions to set blend modes, and draw rectangles.

Establish a coordinate system Draw vectors to represent the velocities of the boat relative to the water and the marble relative to the boat Known:

Unknown:

Because the two velocities are at right angles, use the Pythagorean theorem vm/w2 vb/w2 vm/b2 vm/w vb/w2 41 m/s Find the angle of the marble s motion tan tan

vm/b2 (075 m/s)2

(40 m/s)2

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Start by creating a new application in Xcode. The new application will use both NSDocument architecture and Core Data, so you ll want to pick the appropriate template when creating the application. If you re running Xcode 3.1 on Leopard, this means choosing the Core Data Document-based Application template in the Mac OS X / Application section of the New Project assistant; in Snow Leopard, it means choosing the Cocoa Application template, then making sure that the checkboxes for creating a document-based application and for using Core Data are both checked. Name the new

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Are the units correct Dimensional analysis verifies that the velocity is in m/s Do the signs make sense The signs should all be positive Are the magnitudes realistic The resulting velocity is of the same order of magnitude as the velocities given in the problem

22 You are riding in a bus moving slowly through heavy traffic at 20 m/s You hurry to the front of the bus at 40 m/s relative to the bus What is your speed relative to the street 23 Rafi is pulling a toy wagon through the neighborhood at a speed of 075 m/s A caterpillar in the wagon is crawling toward the rear of the wagon at a rate of 20 cm/s What is the caterpillar s velocity relative to the ground 24 A boat is rowed directly upriver at a speed of 25 m/s relative to the water Viewers on the shore see that the boat is moving at only 05 m/s relative to the shore What is the speed of the river Is it moving with or against the boat 25 An airplane flies due north at 150 km/h relative to the air There is a wind blowing at 75 km/h to the east relative to the ground What is the plane s speed relative to the ground

Another example of combined relative velocities is the navigation of migrating neotropical songbirds In addition to knowing in which direction to fly, a bird must account for its speed relative to the air and its direction relative to the ground If a bird tries to fly over the Gulf of Mexico into too strong a headwind, it will run out of energy before it reaches the other shore and will perish Similarly, the bird must account for crosswinds or it will not reach its destination You can add relative velocities even if they are at arbitrary angles by using the graphical methods that you learned in 5 Remember that the key to properly analyzing a two-dimensional relative-velocity situation is drawing the proper triangle to represent the three velocities Once you have this triangle, you simply apply your knowledge of vector addition from 5 If the situation contains two velocities that are perpendicular to each other, you can find the third by applying the Pythagorean theorem; however, if the situation has no right angles, you will need to use one or both of the laws of sines and cosines

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