# Kinematics in Two Dimensions (PowerPoint)

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```					   Chapter 3

Projectile Motion

Alta High Conceptual Physics
Vectors and Scalars
   Vector vs. Scalars
   Vectors have both magnitude and direction
   Displacement, Velocity, Acceleration, Force,
and Momentum
   Scalars have only magnitude
   Mass, Time, and Temperature

Alta High Conceptual Physics
   Two Methods
   Graphical Method
   Requires a ruler and a protractor
   Process
   Convert each vector into a line that fits a scale of
   Draw Vectors head to tail, measuring the angle
exactly
   Draw a resultant
   Measure the resultant and the resulting angle
   Convert the measurement back into a vector answer

Alta High Conceptual Physics
A man drives 125 km west and then turns
45º North of West and travels an additional
100 km. What is his displacement?
Step one: Set Scale
1 cm = 25 Km

100 km = 4 cm

45°

Alta High 5 cm
125 km = Conceptual Physics
Step Two: Move Vectors Head to Tail
Step Three: Draw Resultant from Tail of First
Vector to the head of the last.
100 km = 4 cm

45°

125 km = 5 cm

Alta High Conceptual Physics
Step Four:    Measure the Measure the Resultant
and the resulting angle.
Step Five:    Using the Scale, convert the
100 km = 4 cm       measurement into an answer.

8.3 cm
8.3 cm * 25 Km/cm = 208 Km
45°

125 km = 5 cm         26.2°

Final Answer: His displacement is 208
of West
Km @ 19.9° North High Conceptual Physics
Alta
Components of Vectors
   Often we need to change a single
vector into two vectors in order to make
addition easier. These two vectors will
be a right angles to each other and
are called component vectors or just
components for short.
   The breaking down of a vector this way
is called resolution.
Alta High Conceptual Physics
Vector Resolution
Step One: Draw Original Vector

45º

100 Km 45°
North of West

Alta High Conceptual Physics
Vector Resolution
Step Two: Draw parallelogram to
Resolve Vector into components

100 Km 45°
North of West

Alta High Conceptual Physics
Vector Resolution
Step Three: Label and measure
components
North component = 70.7 km
45°

West component = 70.7 km

100 Km 45°
North of West

Alta High Conceptual Physics
Requires a basic knowledge of
Geometry – You must know:
   Sin θ = Opposite/Hypotenuse
   Tan θ = Sin θ/ Cos θ =Opposite/Adjacent
   ArcSin, ArcCos, ArcTan

Alta High Conceptual Physics
   Mathematical addition requires that you
be able to
   Draw a rough sketch of the original vectors
   Draw a parallelogram
   Draw the Resultant
   Use Triangle geometry to find the
magnitude and direction of the resultant

Alta High Conceptual Physics
Example
What is the result of adding two vectors if
the first vector is 100 km to the west
and the second vector is 50 km to the
north?

Alta High Conceptual Physics
Step One: Sketch the vectors
Step Two: Put vectors on the same axis and draw the
parallelogram

100 Km West
+        50 Km North

Step One                                  Step Two
Alta High Conceptual Physics
Step Three: Draw the Resultant
Step 4: Use Right triangle geometry to find the
length of the resultant

Step Four:
50 Km
Since this is a right triangle you can
find the resultant using Pythagorean
100 Km                     theorem
Pythagorean Theorem: C2 = A2 + B2
111.8 Km              C = SQRT(A2 + B2) =
SQRT{(100)2+(50)2} =111.8 Km

Step Three

Alta High Conceptual Physics
Graphical vs. Mathematical
Methods
   Graphical Method
   Less Complicated Math
   Good Approximation
   Not as exact as you would like
   Mathematical Method
   No need for ruler or protractor
   Less time dedicated to sketch
   More Accurate result
   Mathematics is more difficult
Alta High Conceptual Physics
Projectile Motion
   A projectile is any object thrown or
launched
   Uses the same kinematic equations you
   Requires that you use two sets of
equations; one in the horizontal and
one in the vertical
   These two sets are related only by time

Alta High Conceptual Physics
Projectile Motion
   A projectile which is dropped or thrown
horizontally has an initial velocity in the Y
direction (V0y) = 0
   Acceleration in the horizontal direction (ax) =
0; hence Velocity in the horizontal direction
(Vx) is constant
    This means that only one kinematic equation
applies in the horizontal direction:
Xx = Vxt

Alta High Conceptual Physics
Projectile Motion
   While it is moving forward, it is also falling
just as if you had dropped it.
   So in the Y direction:
x = ½ at2
   Where a is the acceleration due to gravity.
   The one thing that both motions have in
common is time. When the projectile hits the
ground it stops moving in both directions!

Alta High Conceptual Physics
Projectile Motion
A ball is thrown horizontally off a building
which is 200 m high with a velocity of
10 m/s
(a) How long does it take to reach the
ground?
(b) How far from the building will it land?

(c) What is its velocity just before it hits
the ground (remember magnitude and
direction)
Alta High Conceptual Physics
Fast Moving Projectiles:
Satellites
   A satellite is a fast moving projectile
   Since the projectile moves in a curved path,
and the earth is a curved surface, if the
projectile moved fast enough its motion
would follow the curve of the earth
   Such a projectile would fall around the
earth rather than to the earth
   Such a projectile becomes a satellite and
orbits the earth.

Alta High Conceptual Physics
Problem Types
   Resolving vectors into components
   Projectile Motion

Alta High Conceptual Physics

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