Simple Nature - Light and Matter

Solving for the unknowns gives∆x = |∆r| cos θ= 290 km∆y = |∆r| sin θ= 230 kmThe following example shows the correct handling of the plusand minus signs, which is usually the main cause of mistakes bystudents.Negative components example 61⊲ San Diego is 120 km east and 150 km south of Los Angeles. Anairplane pilot is setting course from San Diego to Los Angeles. Atwhat angle should she set her course, measured counterclockwisefrom east, as shown in the figure?⊲ If we make the traditional choice of coordinate axes, with xpointing to the right and y pointing up on the map, then her ∆x isnegative, because her final x value is less than her initial x value.Her ∆y is positive, so we have∆x = −120 km∆y = 150 km .m / Example 61.If we work by analogy with the example 59, we get−1 ∆yθ = tan∆x= tan −1 (−1.25)= −51 ◦ .According to the usual way of defining angles in trigonometry,a negative result means an angle that lies clockwise from the xaxis, which would have her heading for the Baja California. Whatwent wrong? The answer is that when you ask your calculator totake the arctangent of a number, there are always two valid possibilitiesdiffering by 180 ◦ . That is, there are two possible angleswhose tangents equal -1.25:tan 129 ◦ = −1.25tan (−51 ◦ ) = −1.25You calculator doesn’t know which is the correct one, so it justpicks one. In this case, the one it picked was the wrong one, andit was up to you to add 180 ◦ to it to find the right answer.198 Chapter 3 Conservation of Momentum