Begin the
slide show.
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…
dt
4.00 m
2.00 m
…can not be found using right-triangle math because WE
DON’T HAVE A RIGHT TRIANGLE!
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…
We can add the two
individual displacement
vectors together by first
separating them into pieces,
called x- & y-components
This can’t be solved using our right-triangle math because
it isn’t a RIGHT TRIANGLE!
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°
This was the situation...
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…
dt
4.00 m
2.00 m
R1 = 2.00 m, 25° N of E
R2 = 4.00 m, 20° E of N
R1 = 2.00 m, 25° N of E
25°
0.84524 m
1.81262 m
x = R cosθ = (2.00 m) cos 25° = +1.81262 m
y = R sinθ = (2.00 m) sin 25° = +0.84524 m
R2 = 4.00 m, 20° E of N
3.75877 m
θ = 70˚
1.36808 m
x = R cosθ = (4.00 m) cos 70° = +1.36808 m
y = R sinθ = (4.00 m) sin 70° = +3.75877 m
So, you have broken the two individual
displacement vectors into components.
Now we can add the x-components
together to get a TOTAL XCOMPONENT; adding the ycomponents together will likewise give
a TOTAL Y-COMPONENT.
Let’s review first…
R1 = 2.00 m, 25° N of E
25°
0.84524 m
1.81262 m
x = R cosθ = (2.00 m) cos 25° = +1.81262 m
y = R sinθ = (2.00 m) sin 25° = +0.84524 m
R2 = 4.00 m, 20° E of N
3.75877 m
1.36808 m
x = R cosθ = (4.00 m) cos 70° = +1.36808 m
y = R sinθ = (4.00 m) sin 70° = +3.75877 m
We have the following information:
x
y
R1
+1.81262 m
+0.84524 m
R2
+1.36808 m
+3.75877 m
Now we have the following information:
x
y
Adding
the +1.81262
x-components
together and
R1
m +0.84524
m the ycomponents together will produce a TOTAL xand y-component; these are the components of
R2resultant.
+1.36808 m +3.75877 m
the
x
y
R1
+1.81262 m
+0.84524 m
R2
+1.36808 m
+3.75877 m
+3.18070 m
+4.60401 m
x-component of resultant
y-component of resultant
Now that we know the x- and ycomponents of the resultant (the total
displacement of the ant) we can put those
components together to create the actual
displacement vector.
dT
θ
3.18070 m
4.60401 m
The Pythagorean theorem will produce the
magnitude of dT:
c2 = a2 + b2
(dT)2 = (3.18070 m)2 + (4.60401 m)2
dT = 5.59587 m  5.60 m
A trig function will produce the angle, θ:
tan θ = (y/x)
θ = tan-1 (4.60401 m / 3.18070 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.
55°
55º
55º
55º
…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)
dT
4.60401 m
θ
3.18070 m
From the diagram we can see that the angle is
referenced to the +x axis, which we refer to as EAST.
The vector dT is 55° north of the east line; therefore, the
direction of the dT vector would be
55° North of East
So, to
summarize
what we just
did…
We started with the following vector addition situation…
dt
4.00 m
2.00 m
…which did NOT make a right triangle.
Then we broke each of the individual vectors
( the black ones) into x- and y-components…
And now we have a right triangle we can analyze!
dt
…and added them together to get x- and ycomponents for the total displacement vector.
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
(west)
(south)
(east)
(north)
(west)
(continued on next slide)
(south)
(west)
(south)