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Vectors and Projectiles
Vectors and scalars
Vector: A quantity with magnitude (size) direction. Some examples are
displacement, velocity, acceleration, and force.
Scalar: A quantity with magnitude only. Some examples of scalars are
distance, speed, mass, time, and volume.
Vectors are represented by arrows. They can be added by placing the
arrows head to tail. The arrow that extends from the tail of the first vector
to the head of the last vector is called the resultant. It indicates both the
magnitude and direction of the sum.
+
=
or
(resultant)
Remember, vectors don’t always have to be in a straight line but may be
oriented at angles to each other, such as:
or:
Resultant vectors can be determined by a number of different methods.
Here you will solve vector addition exercises both graphically and with
vector components.
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Graphical addition of vectors: Using a ruler, draw all vectors to scale and
connect them head to tail. The resultant is the vector that connects the
tail of the first vector with the head of the last. Graph paper certainly
makes this easier. It honestly does not matter what order you put the
vectors in as long as they are lined up head to tail. For Example:
 Go do your vector assignment while this is fresh in your brain. You
can either ctrl click the link here, or access it through my website
under the vector activity.
http://www.physicsclassroom.com/Physics-Interactives/Vectors-andProjectiles/Name-That-Vector/Name-That-Vector-Interactive
 You WILL need graph paper to show your work! It will help if you
make sure it’s on full screen also.
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Vector Components: because a vector
has both magnitude and direction, you
can separate it into horizontal (or x) and
vertical (or y) components. To do this,
draw a rectangle with a horizontal and
vertical sides and diagonal equal to the
vector. Draw arrow heads on one horizontal
and one vertical side to make the original
vector the resultant of the horizontal and
vertical components.
y
xx
After you have drawn the components, you can find their lengths by using
trigonometry.
In any triangle you can find the angles and lengths using the following
steps.
hyp
opp
ϴ
adj
In this triangle, hyp is the original vector and the hypotenuse of the
triangle
adj is the horizontal component of the vector and also the adjacent
side to the angle
opp is the vertical component of the vector and also the opposite side
to the angle
You can use these equations to solve for any side of the triangle:
sin ϴ = opp
hyp
cos ϴ = adj
hyp
tan ϴ = opp
adj
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For example, if I had the following triangle, I would solve it as follows:
30
?
35°
sin ϴ = opp
hyp
sin 35° = opp
30
To solve for the unknown, you
would multiply by 30, just like a
normal algebra problem.
30 x sin 35° = opp
= 17.2
So if this were a vector problem that read: a force of 30 N was applied to
an object at a 35° angle above the horizontal. What was the amount of
force in the vertical direction?
You would draw this triangle to help solve it and your answer would be
17.2 N
You can also use the “arc” function to solve for an angle of the triangle.
For the following triangle, the functions would be as follows
hyp
opp
ϴ
adj
sin
-1
(opp ÷ hyp) = ϴ
cos
-1
(adj ÷ hyp) = ϴ
tan
-1
(opp ÷ adj) = ϴ
To do the “arc” function, you will push “2nd”
on the calculator, then the trig function.
This will give you the angle.
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To solve the angle on the triangle below, you would do as follows:
35
ϴ
48
tan
-1
(opp ÷ adj) = ϴ
tan
-1
(35 ÷ 48) = ϴ
= 36.1°
If this were a vector problem, it might read: A projectile was launched with
a horizontal speed of 48 m/s and a vertical speed of 35 m/s, at what angle
was the projectile launched?
To solve it, you would draw the triangle above and solve it to get and
angle of 36.1°
 Now you can solve the “triangles practice” included in your packet