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Drawing Physics Pictures
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How to use vectors to solve problems involving displacement,
velocity, acceleration, and force.
A vector quantity is a quantity that has magnitude (size) & direction.
ex. displacement (I traveled 20 m East.)
velocity (The car was going 55 mph at 60o North of East.)
acceleration (The airplane accelerated at 18000 mi/hr2 due South.)
force (The force of gravity was 25 N down.)
A scalar quantity has only magnitude.
ex. distance (I traveled 20 m.)
speed (The car was going 55 mph.)
Vectors can be drawn as arrows – the magnitude represented by the
length and the direction represented by the angle.
ex. 55 mph at 60o North of East (this means N of the E line)
55 mph
60o
Once they are drawn, we can use either geometry or trigonometry to
add them, subtract them, and figure out many things about our
problem.
Graphical Addition of Vectors
Using Geometry
Using this vector strategy, you need to work with a protractor and a
ruler to draw the vector(s) to scale and then measure to find the
answer.
Try this one…
Superman’s girlfriend, Lois, lives 4 miles 45o NE of his “booth”. He
promised he’d sweep her off her feet (literally) at 6:30 pm. It’s now
6:20 pm and his super powers are not functional!
Draw a vector of the displacement to Lois’ house. Use the scale that
one cm equals one mile.
Superman must follow the streets that only run north/south and
east/west instead of leaping over tall buildings in a single bound.
Draw his path along the streets using 2 vectors. Be sure to include
the magnitude and direction of each vector. Does it matter whether
he travels N/S or E/W first?
The 4 mile 45o NE vector (hypotenuse) is called the RESULTANT (the
result).
The other two (legs of the triangle) are called the components (the
pieces).
Practice using graphical methods to solve the following problems.
Use a sheet of graph paper and make your own axes for each
problem.
1. A man lost in a maze makes three consecutive displacements so
that at the end of the walk he is right back where he started. The
first displacement is 8.00 m westward, and the second is 13.0 m
northward. Find the magnitude and direction of the third
displacement, using the graphical method.
2. A dog searching for a bone walks 3.5 m south, then 8.2 m at an
angle 30.0o north of east, and finally 15.0 m west. Find the dog’s
resultant displacement vector using graphical techniques.
3. In order to get from point A to point B, a person follows these
paths:
40 m at 20o N of W
50 m at 20o W of N
Draw a scale diagram of this situation and graphically determine the
resultant displacement (magnitude & direction).
Does it matter what order the vectors are drawn in?
Draw it (on the same diagram) both ways. What kind of geometric
shape do you get?
What does the resultant “do” to the geometric shape?
This is called the ____________________ method of vector addition.
4. In order to get from point A to point B, a person follows these
paths:
30 m at 20o N of E
20 m at 80o S of E
40 m at 30o W of S
Draw a scale diagram of this situation and graphically determine the
resultant displacement (magnitude & direction).
Does it matter what order the vectors are drawn in?
This time there are too many sides to get the same geometric shape
as in #3.
This is called the ____________________ method of vector addition.
Vectors can also be written in ijk format. This is similar to using xyz
coordinates to plot a point. For example, vector A in the following
diagram would be 1i + 3j + 0k, vector B would be 3i + 2j + 0k, and
vector C would be 2i + 0j + 0k. The k component is 0 in all parts
since there is no k (in/out of the page) axis.
y
To add or subtract vectors using
this method, (instead of
parallelogram or tail-to-tip) just
add or subtract the i, j, and k
components. For example,
A + B = 4i + 5j + 0k
A
B
Try A + B + C
C
x
Try A – B
Graphical Addition of Vectors
Using Trigonometry
Using this vector strategy, you need to draw a sketch of the situation
(no need to have it drawn to scale, but approximations help) and
then use the definitions of sin, cos, tan (or the Law of Sines) and the
Pythagorean Theorem to find the answer.
Just to review:
sin  = opposite/hypotenuse
cos  = adjacent/hypotenuse
tan  = opposite/hypotenuse
Law of Sines
a
b
c
-------- = -------- = -------sin A
sin B
sin C
Pythagorean Theorem
a2 + b2 = c2
c
A
b
B
C
a
In this right triangle, c is the hypotenuse. For angle B, b is the
opposite side and a is the adjacent side. For angle A, a is the
opposite side and b is the adjacent side.
Using trigonometry, try the following. Remember to draw a sketch.
Also keep in mind that both magnitude and direction are needed to
define a vector.
1. A golfer takes two putts to get his ball into the hole once he is on
the green. The first putt displaces the ball 6.00 m east, and the
second, 5.40 m south. What displacement would have been needed
to get the ball into the hole on the first putt?
2. A person walks 25.0o north of east for 3.10 km. How far would a
person walk due north and due east to arrive at the same location?