Displacement from Dallas to Chicago

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SUMMARY

The discussion focuses on calculating the displacement from Dallas to Chicago using vector analysis based on given distances and angles. The user initially calculates the components of the vectors from Dallas to Atlanta and from Atlanta to Chicago, resulting in a total displacement magnitude of 1515.4 miles. However, the user is advised to correct the angle measurement for the Chicago vector, which is given from north instead of east, and to apply the cosine rule for accurate vector addition. The importance of treating displacement as a vector quantity is emphasized throughout the discussion.

PREREQUISITES
  • Understanding of basic trigonometry, specifically sine and cosine functions.
  • Familiarity with vector addition and representation in a Cartesian coordinate system.
  • Knowledge of the cosine rule for calculating unknown sides or angles in triangles.
  • Ability to interpret angles measured from different reference directions (e.g., north vs. east).
NEXT STEPS
  • Study vector addition techniques in physics and mathematics.
  • Learn how to apply the cosine rule in various geometric problems.
  • Explore trigonometric functions in depth, focusing on their applications in navigation and mapping.
  • Investigate the implications of modeling the Earth as flat versus spherical in displacement calculations.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are involved in vector analysis, navigation, or geographical calculations will benefit from this discussion.

physics=death
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Hello, I am having some trouble with this question:
A map suggests that Atlanta is 731 mi in a direction of 4.90° north of east from Dallas. The same map shows that Chicago is 561 miles in a direction of 21.1° west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Here's what I did:
731cos(4.90)= 728
731sin(4.90)= 62.4

728+62.4= 790.4

561cos21.1= 523
561sin21.1= 202

523+202= 725

790.4+725= 1515.4 miles<----- magnitude of displacement

I have no idea if I am doing it right...
 
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Your best bet is to draw a triangle and fill it in with the information you have. You're given two of the lengths so all you need to do is find one of the internal angles and then apply the cosine rule.

Remember that on a flat Earth that there are 90 degrees between North and East, East and South and so on.
 
Displacement is a vector. You probably do want to use vectors. So you've got the Dallas-Atlanta vector to be (728,62.4)mi. The first component is EW distance (E positive) and the second is NS distance (N positive). On the second one you have a problem. The angle should be measured counterclockwise from east. You are given the angle from north. Can you fix that? Then how do you combine the two vectors to get Dallas-Chicago?
 

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