Calculating Resultant Displacement in a Cave: A Trigonometry Problem

[Physically Impaired]

Here's the problem:

While exploring a cave a student starts at the entrance and moves the following distances. She goes 75.0 m north, 250 m east, 125 m at an angle 30.0 degrees north of east, and 150 m south. Find the resultant displacement from the cave entrance.

I drew a diagram to make triangles. The first triangle with Y=75 and x=250 gave me 261 m hypotenuse using the pythagorean formula. This was one component of my resultant displacement. I got stuck on the other two. Am I approaching this correctly?

Thanks Bill

 

[arildno]

Welcome to PF!
Have you covered vectors yet?

 

[Physically Impaired]

Yes. We are currently doing projectile motion, but of course I am having trouble. I'm a Bio major with a 3.8 index. I think I just need to be pointed in the right direction (no pun intended). I need to get the appropriate vector neurons to start firing.

 

[HallsofIvy]

Drawing the triangles is good but using the Pythagorean theorem to find the hypotenuse (straight line distance) is not enough by itself because you are not asked about straight line distance. "Displacement" means (as arildno implied) the vector from the starting point to the end point, which might be given as straight line distance with an angle. In any case, you don't want to use "Pythagoras" until you have the line between initial and final position.

Basically, the best thing to do is to break these into components (in "vector" speak) which is really just separating north-south from east-west. That's pretty easy when each motion is given as "north" or "east" or "south". The only complicating part is that "125 m at an angle 30.0 degrees north of east". Draw a picture and use trig to calculate the legs (east and north) of a right triangle with hypotenuse of length 125 m and angle 30 degrees. Combine all the north-south legs (north +, south -) and combine all the east-west legs (east +, west -) to get the legs of your final right triangle.

 

[arildno]

OK, let's use vectors then (south being -north).
Then the first part of the trip gets you to (75,0).
Adding (0,250) to that gets you to (75,250).

Now, traveling 125 meters 30 degrees to the north of east, must be:
125(sin(30),cos(30))
Agree with that?

See what you get out of this technique
(Hint: the last part, going south, can be represented by (-150,0)

 

[Physically Impaired]

Thanks for your help so far. I feel really dumb so please be patient with me.

Ok, I took 250/75 and got the inverse tangent of 73.3 degrees. Then I took the sin of 73.3 degrees (.9578) and divided 250 by .9578 to get vector quantity of 261.

Is this correct so far?

 

[arildno]

What ARE you doing??
1) Determine the vectors you are supposed to add together.
2) Add the vectors together
3. Find the net distance traveled by Pythagoras

 

[HallsofIvy]

So far, you payed NO attention whatsoever to what we've said. The sum of the FIRST TWO moves is completely irrelevant. Find the "east" and "north" distances of the FINAL position and apply Pythogoras and trigonometry to THAT.