Help needed with displacement problem?

Question

appreciate any help solving this problem

A map suggests that Atlanta is 729 mi in a direction of 5.00° north of east from Dallas. The same map shows that Chicago is 559 miles in a direction of 20.8° west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Find magnitude in miles

Find direction in° north of east

Answer 1

I misread it...

Call Dallas (0,0), Chicago is located at [729*cos(5)-559*cos(69.2), 729*sin(5)+559*sin(69.2)] = (527, 586)

Both are to 3 sig. figs.

The distance is from Pythag's thm and is 788 miles

As for angular position, the angle NofE is arctan(586/527) = 48.0 degrees.

-Fred

Note: I was hoping to use the law of cosines, but you need to do the vector stuff first to get the angles and so it winds up being more work, but to check our answer...

C^2 = A^2 +B^2 - 2ABcos(angle) and

C=SQRT(729^2 +559^2 - 2*559*729*cos (74.2))= 788 miles  CORRECT

The angle can be obtained from the law of sines (which I leave to you)...  :)

-Fred

Answer 2

assume Dallas is the coordinate center of this map

call A the vector from Dallas to Atlanta; B the vector from Atlanta to Chicago

vector A can be written as A=729 cos 5 x + 729 sin 5 y

where x and y are unit vectors in the x and y directions

the vector B can be written as

B= - 559 sin20.8x + 559 cos 20.8 y

the vector from Dallas to Chicago is the vector A+B, so add components to get:

A+B=729cos5-559sin20.8 x + 729sin5 + 559 cos 20.8 y

this vector = 527.7 x + 586.1y

the total displacement is the square root of the squares of the components, and is 788.7 mi

the angle can be found from:

tan (theta) = y component/x component = 586.1/527.7



=1.11  or theta=48 degrees

Answer 3

If you use Atlanta as a point (0, 0), you can find out how much in each direction Dallas and Chicago is. Then use the Distance Formula to find the distance between Dallas and Chicago.

First, find the x, y coordinates of Dallas.

Atlanta is 729 mi from Dallas. It is 5.00° North of East.

The x distance from Dallas to Atlanta is 729 cos (5.00°) = 726 mi East

The y distance from Dallas to Atlanta is 729 sin (5.00°) = 63.5 mi North

Therefore, Atlanta is 726 mi East and 63.5 mi North of Dallas.

That makes Dallas 726 mi West and 63.5 mi South of Atlanta:  (-726, -63.5)

Now lets find the x and y coordinates of Chicago in relation to Atlanta.

Chicago is 559 mi from Atlanta in the direction 20.8° West of North, which is 110.8° North of East.

So the x distance is 559 cos (110.8°) = -199 mi East or 199 mi West

And the y distance is 559 sin (110.8°) = 523 mi North

Therefore, Chicago is 199 mi West and 523 mi North of Atlanta. (-199, 523)

Using the distance formula d² = (x2 - x1)² + (y2 - y1)²

d² = (-199 - (-726))² + (523 - (-63.5))²

d² = (726 - 199)² + (523 + 63.5)²

d² = (527)² + (586.5)²

d² = 277729 + 343982.25 = 621711.25

d = √(621711.25) = 788 mi

Chicago is 788 mi from Dallas.

Find the direction:

tan (Ï´) = (y2 - y1) / (x2 - x1)

tan (Ï´) = (523 - (-63.5)) / (-199 - (-726))

tan (Ï´) = 586.5 / 527 = 1.11

ϴ = tan¯¹ (1.11) = 48.1° North of East

So Chicago is 788 mi in the direction 48.1° North of East from Dallas.

Answer 4

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