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

An ant walks 2.00 m 25° N of E, then turns and walks 4.00 m 20° E of N. This can’t be solved using our right-triangle math because it isn’t a RIGHT TRIANGLE! We can add the two individual displacement vectors together by first separating them into pieces, called x- & y-components The total displacement of the ant…

1) A vector with a -x component and a +y component…

2) A vector with a +x component and a - y component…

3) A vector with a +x component and a +y component…

4) A vector with a -x component and a - y component…

5) A vector with a -x component and a zero y component…

6) A vector with a zero x component and a -y component…

7) For the vector 1350 ft, 30° N of E… R = 1350 ft θ = 30°

8) For the vector 14.5 km, 20° W of S… R = 14.5 km θ = 70°

9) For the vector 2400 m, S… R = 2400 m θ = 90°

An ant walks 2.00 m 25° N of E, then turns and walks 4.00 m 20° E of N m 2.00 m dtdt This was the situation... The total displacement of the ant… R 1 = 2.00 m, 25° N of E R 2 = 4.00 m, 20° E of N

R 1 = 2.00 m, 25° N of E 25° x = R cosθ = (2.00 m) cos 25° = m y = R sinθ = (2.00 m) sin 25° = m m m

R 2 = 4.00 m, 20° E of N x = R cosθ = (4.00 m) cos 70° = m y = R sinθ = (4.00 m) sin 70° = m m m θ = 70˚

So, you have broken the two individual displacement vectors into components. Now we can add the x-components together to get a TOTAL X- COMPONENT; adding the y- components together will likewise give a TOTAL Y-COMPONENT. Let’s review first…

R 1 = 2.00 m, 25° N of E 25° x = R cosθ = (2.00 m) cos 25° = m y = R sinθ = (2.00 m) sin 25° = m m m

R 2 = 4.00 m, 20° E of N x = R cosθ = (4.00 m) cos 70° = m y = R sinθ = (4.00 m) sin 70° = m m m

We have the following information: xy R1R2R1R m m m m

Now we have the following information: xy R1R2R1R m m m m Adding the x-components together and the y- components together will produce a TOTAL x- and y-component; these are the components of the resultant.

xy R1R2R1R m m m m m m x-component of resultanty-component of resultant

Now that we know the x- and y- components of the resultant (the total displacement of the ant) we can put those components together to create the actual displacement vector m m dTdT θ

The Pythagorean theorem will produce the magnitude of d T : c 2 = a 2 + b 2 (d T ) 2 = ( m) 2 + ( m) 2 d T = m  5.60 m A trig function will produce the angle, θ: tan θ = (y/x) θ = tan -1 ( m / m) = 55º

Of course, ‘55º’ is an ambiguous direction. Since there are 4 axes on the Cartesian coordinate system, there are 8 possible 55º angles. 55º 55° …and there are 4 others (which I won’t bother to show you). To identify which angle we want, we can use compass directions (N,S,E,W)

m m dTdT θ From the diagram we can see that the angle is referenced to the +x axis, which we refer to as EAST. The vector d T is 55° north of the east line; therefore, the direction of the d T vector would be 55° North of East

So, to summarize what we just did…

We started with the following vector addition situation… 4.00 m 2.00 m dtdt …which did NOT make a right triangle.

dtdt Then we broke each of the individual vectors ( the black ones) into x- and y-components… …and added them together to get x- and y- components for the total displacement vector. And now we have a right triangle we can analyze!

Yeah, baby! Let’s give it a try! Complete #16 on your worksheet. (Check back here for the solution to the problem when you are finished.)

# 16 (continued on next slide) (west) (south) (east) (north) (west)(south)

(west)(south)