Transcript Notes: Vectors Part 3

Method #3:
Law of Sines and Cosines
Also referred to as the analytical
method.
Steps


Draw a rough sketch of the vectors
The resultant is determined using:



Algebra
Trigonometry
Geometry
These Laws Work for
Any Triangle.
A + B + C = 180°
C
Law of sines:
a
sin A
b
a
=
b = c
sin B
sin C
Law of cosines:
B
A
c
c2 = a2 + b2 –2abcos C
Example 2: using method 3
Stan is trying to rescue Kyle from drowning. Stan
gets in a boat and travels at 6 m/s at 20o N of E, but
there is a current of 4 m/s in the direction of 20o E
of N. Find the velocity of the boat.
Example (using same problem)
Stan is trying to rescue Kyle from drowning. Stan
gets in a boat and travels at 6 m/s at 20o N of E, but
there is a current of 4 m/s in the direction of 20o E
of N. Find the velocity of the boat.
Calculating:
Magnitude:
c2
= a2 + b2 – 2abcosC
= (6m/s)2 + (4m/s)2 – 2(6m/s)(4m/s)cos130°
= 82.85
c = 9.10 m/s
Direction:
sin C
=
c
sin 130°
=
9.10
sin B = 0.337
R = 19.67° + 20°
R = 9.1 m/s @ 39.7° N of E
sin B
b
sin B
4
B = 19.67°
= 39.67°
Use the Law of:

Sines when you
know:


2 angles and an
opposite side
2 sides and an
opposite angle

Cosines when
you know:

2 sides and the
angle between
them
Advantages and Disadvantages
of the Analytical Method



Does not require
drawing to scale.
More precise
answers are
calculated.
Works for any
type of triangle
if appropriate
laws are used.



Can only add 2
vectors at a
time.
Must know many
mathematical
formulas.
Can be quite
time consuming.
This completes Method Three!
Keep up the good work! This is our
last time in class to learn these.
problems #5, 6 due tomorrow
Another Problem
Paul is on a railroad flat car
which is moving east at 20.0
m/s (Vcg = velocity of the car
relative to the ground) . Paul
walks on the flat car at 5.0 m/s
@ 40.0o N of E as shown (Vpc
= velocity of Paul relative to
the car) . What is Paul’s
velocity relative to the ground
(Vpg = velocity of Paul relative
to the ground)?
Vpg = 24 m/s @ 7.7o (or 7.7o N of E)