Physics Question: Kinematics: Please let me know how to solve it. Thanks?

Question

A man entered a cave and walked 100m north. He then made a sharp turn 150 deg to the west and walked 87m straight ahead. How far is the man from where he entered the cave?

the answer is 50m

Answer 1

if you make the man's initial trajectory as being straight and then created another triangle off of it (it's kind of hard to visualize) then you can use sines and cosines to figure out how far he is. which is why i like to draw pictures :)


so to find the angle, the hypotenuse of the smaller triangle is 87m. we already know the angle to be 30* (180-150).
so sin(30) = x/87 because sin(angle)=opp/hyp
so if we solve for x, 87*sin(30)=x=43.5
we can then use the pythagorean theorem to find the length of the remaining side. so 87^2-43.5^2=side^2
so side~75.34m

we can subtract this from 100 and again use the pythagorean theorem to find the distance from the opening.
so the distance would be our c side. so c^2=75.34^2+43.5^2
c^2=5676.75+1892.25
c^2=7569
c=distance from cave mouth = 87 m

i hope my math was right :)

Answer 2

This is not a kinematics question (kinematics deals with the relationships between displacement, velocity and acceleration) -- it's really a trigonometry question. It's useful to draw a diagram. Draw a line going "north" (toward the top of the paper) and label it "100". Then make a turn toward the "west" (the left) at 150 degrees, and draw another line labeled "87" (of course it should be a little shorter than the "100" line). How sharp a turn is that? Hint: if you turned 180 degrees, you'd be heading directly back south again. But your turn is not quite 180. So your "87" line does not go quite south. How far off from south does it point? You should be able to figure that out by subtracting two numbers I've already given you. Figure out that angle, and write it on the diagram, where the two legs meet.

Now your diagram should look sort of like an upside-down "V", with one leg a little longer than the other. The tip of the right leg is where you started, and the tip of the left leg is where you've ended. So the question is: how far is one tip from the other tip? Draw a line from one tip to the other tip, and label it "x", because that's the number you're trying to find.

Now you have a triangle. You know the length of two sides; and you know one of the angles. In theory, that's enough information for you to figure out the third side, using something in trigonometry called the Law of Cosines.

However, there's an easier way in this special case. I think your teacher chose the numbers "100" and "87" and "150" in particular, to make the problem easier to solve.

To see what I mean, draw another diagram. This one should be an equilateral triangle, with each side equal to "100". Make one side horizontal (the "base"), and the other two sides meeting up at the top. Next, draw a vertcal line from the top of the triangle, straight down to the base. If you drew it well, the vertical line should hit the base right in the middle.

Now notice that, in your second diagram, you have split the equilateral triangle into two right triangles. Now use the pythagorean theorem to figure out how long the vertical line is (hint: it's part of a right triangle where the hypotenuse is 100 and one leg is 1/2 of 100).

Now compare the right triangles you drew in your 2nd diagram, to the triangle in your FIRST diagram. Notice any similarities between the angles and the lengths of the sides?

Those are all the hints I'm giving you. If you are not familiar with how to use the pythagorean theorem, then you are definitely not ready for this problem, and you should review.