Raymond A. Serway
John W. Jewett
Chapter 3
Motion in Two Dimensions

3.1 The Position, Velocity, and
Acceleration Vectors
•
The position of an
object is described
by its position vector
•
The
displacement
of the object is the
difference between
its final and initial
positions:

3.1 The Position, Velocity, and
Acceleration Vectors
•
The average velocity is
the ratio of the
displacement to the
time interval for the
displacement:
•
The direction of the
average velocity is the
direction of the
displacement vector

3.1 The Position, Velocity, and
Acceleration Vectors
•
The average velocity between points is
independent of the path
taken
•
This is because it is dependent on the
displacement, which is also independent of the
path
•
If a particle starts its motion at some point and
returns to this point via any path, its average
velocity is zero for this trip since its displacement
is zero

3.1 The Position, Velocity, and
Acceleration Vectors
•
The instantaneous velocity is the limit of the average
velocity as ∆
t
approaches zero:
•
The direction of the instantaneous velocity vector at
any point in a particle’s path is along a line tangent
to the path at that point and in the direction of
motion
•
The magnitude of the instantaneous velocity vector is the
speed

3.1 The Position, Velocity, and
Acceleration Vectors
•
The average acceleration of a particle is the
ratio of the change in the instantaneous
velocity to the time interval:

3.1 The Position, Velocity, and
Acceleration Vectors
•
As a particle moves,
v
can be found in
different ways, according to the rules of
vector addition
•
The average acceleration is a vector quantity
directed along
v

3.1 The Position, Velocity, and
Acceleration Vectors
•
The instantaneous acceleration is the limit of
the average acceleration as
v
/
t
approaches
zero:

3.1 The Position, Velocity, and
Acceleration Vectors
•
Various changes in a particle’s motion may
produce an acceleration
•
The magnitude of the velocity vector may change
•
The direction of the velocity vector may change
•
Even if the magnitude remains constant
•
Both may change simultaneously

3.2 Two-Dimensional Motion with
Constant Acceleration
•
When the two-dimensional motion has a
constant acceleration, a series of equations
can be developed that describe the motion
•
These equations will be similar to those of one-
dimensional kinematics
•
Motion in two dimensions can be modeled as
two
independent
motions in the
x
and
y
directions
•
Any influence in the
y
direction does not affect the
motion in the
x
direction, and vise versa

3.2 Two-Dimensional Motion with
Constant Acceleration
•
Position vector:
•
Velocity:
•
Since acceleration is constant, we can also find
an expression for the velocity as a function of
time:
•
The position vector as a function of time can be
written as

3.2 Two-Dimensional Motion with
Constant Acceleration
•
The velocity and position vectors can be
represented by components:

Example 3.1 Motion in a Plane

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- Spring '17
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