Displacement, Velocity, and Acceleration
Equations of Kinematics in 2 Dimensions
Projectile Motion
Relative Velocity
The displacement vector of an object is drawn
from the initial position vector to the final
position vector. The magnitude of the initial
plus the displacement will be the final position
vector:
r
f
= r
0
+ r
Therefore, the displacement vector can be
found by finding the difference.
r =
r
f
–
r
0
See figure 3.1 pg 59
The velocity and acceleration calculations do
not change in two dimensions. It is simply
important to remember 2 things:
The x and y components must be treated
individually and then added together
The direction is important since these are vector
quantities. A change in direction will result in an
acceleration, even if the magnitude of the velocity is
not changing.
Ex: Check for understanding 1, pg 60,
Pg 82 # 3, 7,9
If an object moves in the horizontal and vertical
direction at the same time, we assign horizontal
motion with an x (
v
x
, a
x
), and vertical motion
with a y (
v
y
, a
y
)
See table 3.1 page 61
!! It is important to realize that the x part of the
motion occurs exactly as it would if the y part
did not occur at all. Similarly, the y part of the
motion occurs exactly as it would if the x part
of the motion did not exist.
Pg 65 #2, pg82

83 #15, 17, 19
Projectile motion results when an object is thrown
either horizontally through the air or at an angle
relative to the ground. This will result in the object
moving through the air with a constant horizontal
velocity while falling freely under the influence of
gravity. The resulting path of the projectile is
called a trajectory and has a parabolic shape.
Motion of a projectile is broken down into constant
velocity and zero acceleration in the horizontal
direction and changing vertical velocity due to
acceleration of gravity in the vertical direction.
Horizontal Motion
Vertical motion
A
x
= 0
a
y
= g =

9.81 m/s
2
V
x
= x/t
v
y
=
v
oy
+ g t
X =
v
x
t
y =
v
oy
t + ½ g t
2
*notice the minus sign in the equations for vertical motion.
Since the acceleration g and the initial vertical velocity are
in opposite directions, we must give one of them a
negative sign. Horizontal velocity of projectiles is
constant, vertical velocity is changed by gravity
Pg 82 # 21, 27, 29, 33, 43
When adding vectors of one object’s motion
relative to another, we use the ordering of the
subscript symbols in a definite pattern. The first
subscript refers to the body that is moving while
the second letter indicates the object relative to
which the velocity is measured.
Ex.
V
pt
where p is the passenger and t is a train that the
passenger has a specific velocity relative to the train. See
example pg 74.
This can help add vectors to determine relative
velocity of other objects with problems that have
multiple frames of reference. By definition,
V
pt
=

V
tp
Pg 81 #14, 15, pg 85 #55, 57, 61
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