Yea I know, specific title. xD I need help in the Physics textbook (looks like t

Question

Yea I know, specific title. xD I need help in the Physics textbook (looks like the one below) on problem three of the practice set C of chapter three. The problem says: During a rodeo, a clown runs 8m north, turns 55o north of east, and runs 3.5 m. Then, after waiting for the bull to come near, the clown turns due east and runs 5.0m to exit the arena. What is the clown's total displacement?

I've been trying different equations (Sin, Cos, Tan) but I still can't do it. Could someone help and explain how to do this one, please? A drawing would also be super.

Thanks!

Explanation / Answer

Choose which directions you want to label north, south, east and west. I’m going to set up as north, right as east. Ok. Start at some arbitary point. The clown runs 8 meters north; draw a line segment straight up, and label it 8 m. This is a vector of 8 m, direction north. Next, the clown turns 55 degrees north of east. East will be a horizontal line perpendicular to the line segment 8 m. (Imagine a compass, where East is always perpendicular to North.) From that horizontal line, measure a theta of 55 degrees, then draw a line segment 3.5 m of angle theta. This is a vector of 3.5 m, northeast. Lastly, the clown runs east, which will again be a horizontal line. Basically, clown runs to the run 3.5 m. You need to break one of these vectors into components, and it’s going to be your northeast vector. The reason you do this is that it makes adding all the vectors together much, much easier. To break a vector into components, you need to construct a right triangle then compute the x-component and y-component using trigonometric identities. In this case, your angle is 55 degrees, and your HYPOTENUSE is 3.5 m. The side opposite of your angle will be your y-component; the remaining side will be your x-component. To find your y-component, do 3.5*sin(55), and to find your x-component, do 3.5*cos(55). Now that you have your y-component, realize that your y-component is poiting upwards. Add it to your north vector of 8 meters. You’re going to end up with 3.5*sin(55) + 8 meters. Your x-component is pointing to the right; add it to your east vector of 5.0 meters, so that you end up with 3.5*cos(55) + 5.0 meters. Now you have two longer legs of a bigger triangle. Your vector 3.5*sin(55) + 8 meters is the leg pointing up; your vector 3.5*cos(55) + 5.0 meters is pointing right. So how do you find the displacement? The displacement will be the hypotenuse of this bigger triangle. To find that, you need to use the Pythagorean Theorem a^2 + b^2 = c^2 where c is your hypotenuse. You’re going to end up with the square root of [(3.5*sin(55) + 8 meters)^2] + [(3.5*cos(55) + 5.0 meters)^2].

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