This document provides an overview of two-dimensional kinematics. It discusses key concepts such as reference frames, vectors, vector addition and subtraction using graphical and analytical methods, projectile motion, and calculating maximum height and range. The document is based on an open educational resource on college physics and is licensed under Creative Commons.

1. This work is licensed under a Creative Commons Attribution 4.0 International License.

2. 2
Module Overview
Two-Dimensional Kinematics

3. 3
Acknowledgments
This presentation is based on and includes content derived from the
following OER resource:
College Physics 1
An OpenStax book used for this course may be downloaded for free at:
https://openstax.org/details/books/College-Physics

4. 4
Motion in Two Dimensions
Motion that is confined to a flat surface or a plane can be described
using two-dimensional kinematics.
Examples of two-dimensional motion include the flight of a cannonball
through the air, and a group of hockey players skating on an ice rink.
Two-dimensional kinematics can be thought of as an extension of the
one-dimensional kinematics used for describing motion confined to a
straight line, with the provision that two parameters are required to
describe the positions and movements of objects instead of just a
single parameter.

5. 5
Reference Frames for Two Dimensions
A two-dimensional reference frame is most often based on a pair of
perpendicular axes. The two axes define a plane and meet at a common
origin. The axes are usually graduated, meaning that they are labeled
with evenly spaced markings used to indicate the distance from the
origin to an object, as measured along that particular axis.
The two perpendicular axes are most often labeled x and y, which
identify the horizontal axis and the vertical axis, respectively.
The position of each object in two dimensions is identified by specifying
a unique (x, y) pair.

6. 6
Calculating the Length of a Line Segment
The length of a line segment in two
dimensions can be calculated using
the Pythagorean theorem.
A line segment of length c is related to
the sides a and b of the right triangle
depicted in the diagram by the
equation
c2 = a2 + b2.

7. 7
Vectors in Two Dimensions
A vector confined to a single
dimension can point in only one of
two directions. In contrast, two-
dimensional vectors can exhibit an
unlimited number of directions
within a plane.
The two-dimensional vector D
depicted at right possesses a
length, D, and a direction indicated
by the angle θ.

8. 8
Equal Vectors
If a vector is moved to another place in
the x-y plane, while retaining its
direction and original length (or
magnitude), it is still the same vector.
Stated another way, if two vectors have
the same length and the same
direction, they are equivalent.
The diagram to the right indicates that
the vector D has been shifted from the
origin of the x-y coordinate system.
Although it has moved, it is still the
same vector.

9. 9
Negatives of Vectors
The negative of vector B is a vector of equal length but opposite direction
and is denoted –B.
Image: College Physics. OpenStax.

10. 10
Vector Arithmetic
Just as whole numbers, rational numbers, and real numbers can be
added or subtracted from one another, vectors can also be added and
subtracted.
Since vectors possess both a magnitude and a direction, the operations
of addition and subtraction are somewhat more involved than they are
with ordinary numbers.
There are two methods for adding and subtracting vectors: the graphical
method and the analytical method. Although both methods produce
the same result, one method or the other may be chosen depending
upon the particular situation.

11. 11
Adding and Subtracting Vectors Graphically
Adding one vector to another using a graph involves the so-called head-
to-tail method. The head of a vector is indicated by its arrow, while its
tail is situated at the opposite end.
The head-to-tail method involves drawing the first vector on a graph and
then placing the tail of each subsequent vector at the head of the
previous vector. The resultant vector is then drawn from the tail of the
first vector to the head of the final vector.
To subtract one vector from another, for instance vector B from vector
A, one simply draws the negative vector –B and adds it to A.

12. 12
Adding Vectors Using the Head-to-Tail Method
The addition of the three vectors A, B, and C is accomplished by placing
all three on a graph in the manner shown below. The resultant vector R is
obtained by drawing from the x-y origin to the head of vector C. By using
a ruler and a protractor, the magnitude and direction of R can be
manually determined.
Image: College Physics. OpenStax.

13. 13
Adding and Subtracting Vectors Analytically
The analytical method of adding and subtracting vectors means that
equations are solved in order to numerically determine the magnitude
and direction of a resultant vector.
The magnitude of the resultant vector is found by use of the
Pythagorean theorem.
On the other hand, trigonometric identities are used to determine the
direction of the resultant vector, as indicated by an angle the vector
makes with an axis of the x-y coordinate system.

14. 14
Horizontal and Vertical Components of Vectors
The illustration at right shows how
a vector—in this case vector A—
equals the sum of its horizontal and
vertical components along the x
and y axes.
The equations in the diagram
indicate how the two components
of vector A are related to the angle
θ formed by the horizontal axis and
vector A.
Image: College Physics. OpenStax.

15. 15
Using the Sine and Cosine Functions
When adding the vectors A and B together to form a resultant vector R,
the sine (sin) and cosine (cos) trigonometric functions are first used to
obtain the individual horizontal and vertical components of A and B. The
angle θ represents the angle of each vector relative to the x axis.

16. 16
Solving for Magnitude and Direction
Once the values of the horizontal and vertical components of A and B are
determined—that is, the values of Ax and Ay, and Bx and By—these values
are then used to calculate the horizontal and vertical components of R,
namely Rx and Ry. The magnitude and direction of R, denoted R and θ
respectively, can also be calculated. The pertinent equations are:

17. 17
Independent Motion Along Perpendicular Axes
Motion in the horizontal direction
does not affect motion in the
vertical direction, and vice versa.
The illustration at right shows two
balls dropped at the same instant.
The red ball has no initial horizontal
motion, while the blue ball does.
The vertical speeds of both balls
increase at the same rate as they
fall. However, note that the
horizontal motions of both balls
remain unchanged.
Image: College Physics. OpenStax.

18. 18
Projectile Motion
Projectile motion refers to the motion of an object subject only to the
acceleration of gravity, with negligible effects caused by air resistance.
In this scenario (where air resistance is negligible), the object’s horizontal
velocity vx remains constant, expressed mathematically as vx = constant.
In contrast, the object’s vertical velocity vy does change over time due to
the downward acceleration of gravity, expressed as ay = -g = -9.90 m/sec2.
The kinematic equations for vertical displacement (y), and velocity, (vy)
are:

19. 19
Maximum Height and Range of a Projectile
From the kinematic equations for the horizontal and vertical displacement
and velocity, one may calculate the maximum height h (equivalent to the
maximum vertical displacement) and the maximum range R (equivalent to
the maximum horizontal displacement) that a projectile can attain. The
equations for R and h are:

20. 20
How to Study this Module
• Read the syllabus or schedule of assignments regularly.
• Understand key terms; look up and define all unfamiliar words and
terms.
• Take notes on your readings, assigned media, and lectures.
• As appropriate, work all questions and/or problems assigned and as
many additional questions and/or problems as possible.
• Discuss topics with classmates.
• Frequently review your notes. Make flow charts and outlines from
your notes to help you study for assessments.
• Complete all course assessments.

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