Calculating Total Displacement of a Car and Finding Direction in Vector Homework

[courtrigrad]

A car is driven eastward for a distance of 50 miles, then northward for 30 miles, and then in a direction 30 degrees east of north for 25 miles. Find the total displacement of the car from its starting point.

So I set up a vector diagram, starting at the origin. I did: 30^2 + 50^2 and took the square root of that to find the length of the first part. Then I added that to 25 and got 83.234, but the answer is 81.0. What am I doing wrong? Also how would you find the direction?

Thanks

 

[HallsofIvy]

What you are doing wrong is not recognizing that "displacement" is not the same as "distance traveled". If the car traveled 50 miles due east, then turned around and traveled 50 miles due west, it would have traveled 100 miles- but it certainly wouldn't be 50 miles from it's starting point, which is what "displacement" means!

Since you mention the car starts "at the origin", I presume that you want to use the vector components or "coordinates". Okay, setting up a coordinate system so that the unit vector i is East and unit vector j is west, the first vector is 50i (50 miles east), the second is 30j (30 miles north), and the third is 25 cos(30)i+ 25 sin(30)j.
Add those three vectors to find the final position of the car and use the Pythagorean theorem to find the distance that point is from the origin.

 

[courtrigrad]

i still get 83.3 while the answer in the back says 81.0. Originally I did 30^{2} + 50^{2} and took \sqrt{3400} which is 58.31. Then I added that to 25 and got 83.3! Also how would i finf the final direction?

thanks

 

[Päällikkö]

It's wrong to add the 25, as it's not in the same direction as the \sqrt{30^2+50^2}

As HallsofIvy said:
\sqrt{(30+25cos(30))^2+(50+25sin(30))^2}

Compute the total displacement in the east-west direction, and the total displacement in the north-south direction. Separately. Then add 'em together with the Pythagorean theorem.

 

[Pyrrhus]

The answer is correct is 81, the problem is you think the angle made by the first part is 60 degrees with the horizontal or 30 degrees with the vertical which is not the case, it's 30.96 degrees, that's why you find the slight difference between 81 and 83. Check Päällikkö's work for details.