Use either trigonometry or components to find displacement

[aurorabrv]

Homework Statement

Bob walks 160 m south, then jogs 630 m southwest, then walks 160 m in a direction 32degrees east of north.

a) Use either trigonometry or components to find the displacement that will return Bob to his starting point by the most direct route. Give your answer as a distance.

b)Find the direction of the displacement that will return Bob to his starting point by the most direct route.

Homework Equations

Not sure what to use.

The Attempt at a Solution

I'm not even sure where to begin, even some hints to tell me how to start would be very useful!

a)
60 + 630(cos45+sin45) + 160(cos32,sin32)
60 + 630(0.71 + 0.71) + 160(0.848 + 0.5299)
60 + 447.3 + 447.3 + 135.688 + 84.787
642.98 + 533.087

642.98^2 + 533.087^2
413423.28 + 284181.83
697605
take root = 835.22

 

[buffordboy23]

Draw a picture of the displacement vectors. Bob can only move in two dimensions (a plane), so determine the x- and y-components of each vector. Now add the individual components together and you get Bob's net displacement from his initial position. It appears that you did this but I can't tell what the x- and y-components are from your notation. Also I see a 60 as the first time in your work, but not in the problem statement; should it be 160?

The most direct route is straight line, so for part a) apply the Pythagorean theorem to your net x- and y-components and take the square root to get the distance.

 

[baba89]

m

b)
To find the direction, we can use the inverse tangent function to find the angle of the displacement. We can start by finding the x and y components of the displacement:

x = 630(cos45) + 160(cos32)
x = 444.21 + 135.688
x = 579.898

y = 630(sin45) + 160(sin32)
y = 444.21 + 84.787
y = 528.997

Now we can use the inverse tangent function to find the angle:

tan^-1 (528.997/579.898) = 43.46 degrees

Therefore, the direction of the displacement is 43.46 degrees east of north.