Non Right Triangle Vector Addition. Sections 1.7 and 1.8

Transcription

1 Non Right Triangle Vector Addition Sections 1.7 and 1.8

2 Question: Why in the name of all that is good would someone want to do something like THAT? Answer: Because there is no law that states vectors must add up to make right triangles. (Oh, but if only there were.)

3 CONSIDER THE FOLLOWING... An ant walks 2.00 m 25 N of E, then turns and walks 4.00 m 20 E of N. The total displacement of the ant d t 4.00 m 2.00 m can not be found using right-triangle math because WE DON T HAVE A RIGHT TRIANGLE!

4 We can add the two individual displacement vectors together by first separating them into pieces, called x- & y-components

5 Into WHAT????????? COMPONENTS Every vector can be thought of as pointing somewhat horizontally. [This is the black vector s shadow on the y-axis] [This is the black vector s shadow on the x-axis] and somewhat vertically. They re kind of like the vector s shadows.

6 If we add the x- and y-components together they create the original vector and it makes a right triangle!

7 Just a few things to keep in mind... Since X-component vectors can point either EAST or WEST EAST is considered positive. WEST is considered negative.

8 Who's Law Is It, Anyway? Murphy's Law: Anything that can possibly go wrong, will go wrong (at the worst possible moment). Cole's Law?? Finely chopped cabbage 8

9 Law of Sines Let ABC be any triangle with a, b and c representing the measures of the sides opposite the angles with measures A, B, and C respectively. Then c A b B a C sin A sin B sin C = = a b c Law of Sines can be used to find missing parts of triangles that are not right triangles Case 1: measures of two angles and any side of the triangle (AAS or ASA) Case 2: measures of two sides and an angle opposite one of the known sides of the triangle (SSA)

10 The Law Cosines Now use it to solve the triangle below. Label sides and angles Side c first 15 C = A B c c b a 2 a b cosc c cos 26 10

11 Applying the Cosine Law c 15 2 C = cos 26 o 15 C Now calculate the angles using the Law of Sines. A c = 6.65 B sin A sin B sin 26 0 = =

12 Wonder Woman Jet Problem Suppose Wonder Woman is flying her invisible jet. Her onboard controls display a velocity of 304 mph 10 E of N. A wind blows at 195 mph in the direction of 32 N of E. What is her velocity with respect to Aqua Man, who is resting poolside down on the ground? v WA = velocity of Wonder Woman with respect to the air v AG = velocity of the air with respect to the ground (and Aqua Man) v WG = velocity of Wonder Woman with respect to the ground (and Aqua Man)

13 We know the first two vectors; we need to find the third. First we ll find it using the laws of sines & cosines, then we ll check the result using components. Either way, we need to make a vector diagram.

14 v WG v WG 10 v WA + v AG = v WG 80 The 80 angle at the lower right is the complement of the 10 angle. The two 80 angles are alternate interior. The 100 angle is the supplement of the 80 angle. Now we know the angle between red and blue is 132.

15 The law of cosines says: v 2 = (304) 2 + (195) 2-2 (304) (195) cos 132 So, v = 458 mph. Note that the last term above appears negative, but it s really positive, since cos 132 < 0. The law of sines says: 132 sin 132 sin = v 195 So, sin = 195 sin 132 / 458, and v This means the angle between green and the horizontal is Therefore, from Aqua Man s perspective, Wonder Woman is flying at 458 mph at 61.6 N of E.

16 Wonder Woman Problem: Component Method This time we ll add vectors via components as we ve done before. Note that because of the angles given here, we use cosine for the vertical comp. of red but sine vertical comp. of blue. All units are mph

17 Combine vertical & horiz. comps. separately and use Pythagorian theorem. = tan -1 ( / ) = is measured from the vertical, which is why it s 10 more than was mph

18 Comparison of Methods We ended up with same result for Wonder Woman doing it in two different ways. Each way requires some work. You can only use the laws of sines & cosines if: you re dealing with exactly 3 vectors. (If you re adding three vectors, the resultant makes 4, and this method won t work the vectors form a triangle. Regardless of the method, draw a vector diagram!

19 Assignment Non Right Angle Vector Addition Ch. 1 Pages 24-25, Problems 26, 29, 31, 32, 45, 46, 51, 52.