Republic of the Philippines
Department of Education Regional Office IX, Zamboanga Peninsula
9 Zest for Progress
Zeal of Partnership
Science Grade 9
Quarter 4 - Module 2 Motion in Two Dimensions
Name of Learner:
Grade & Section:
Name of School:
Module 2
Motion in Two Dimensions
What I Need to Know
Hello, science enthusiasts! Have you been well? This module is written with you in
mind. It is here to assist you in investigating the relationship between the angle of release and
the projectile's height and range (S9FE-IVa-35).
The scope of this module allows it to be used in many alternative learning situations.
The language uses and recognizes the various terminology level of students. The lessons are set
to follow the standard sequence of the course
After going through this module, you're expected to:
1. Describe the path of an object in projectile motion;
2. Differentiate the horizontal and vertical motions of a projectile; and
3. Explain the relationship between the angle of release and the height and range of the
projectile.
This lesson will discuss a type of motion in two-dimensions using projectile motion as an
example. It focuses on the concept that two-dimension movements will be described and
predicted using kinematics and dynamics. It also illustrates that true
projectiles follow a parabolic path that is due to the downward pull of gravity. The activities
also exhibit that the uniform horizontal motion (non-accelerated) is independent of the non-
uniform (uniformly accelerated) vertical motion.
What's In
What is your favorite sport? Is it basketball, volleyball, badminton, or perhaps, ping
pong? Have you ever joined Intramurals? When you throw a ball, how far can it travel? Or
better yet, how hard do you need to serve for the volleyball to reach the other side of the court?
Whether you're an athlete or a member of the cheering squad, you might have observed that
the ball seems to be "flying" when thrown mid-air and appears to follow a specific path.
Not only that, but you may have noticed that in many sports and games, players come
"flying" too. These situations happen in real life and not only apply to sports but can also be
used to track the path of meteorites and rockets' trajectories. How cool is that?
In this module, you will be introduced to the concepts of understanding motion in two-
dimensions that will help you employ the physics of sports and improve game events
experiences.
What's New
Activity 1. Curve Me on an Incline
Objective: The students will be able to capture a full trajectory of projectile motion
on an inclined surface.
Materials Needed:
Marble or jackstone fine powder (face powder, corn starch)
¼ illustration board half-protractor template
stack of books
4 sheets of graphing paper
set of weights retractable ball pen as launcher
2 popsicle sticks masking tape
tabletop stopwatch
sticky tape
Procedure:
Day One Activity
I. Linear horizontal motion
Instruction: Use the pen to move the marble horizontally along the tabletop. Observe the
ball's motion. Draw and label the velocity-time and the acceleration-time graphs on the axes
below.
Complete the sentence by encircling the answer.
A ball rolling horizontally has a velocity that is (changing, constant) and an
acceleration that is (zero, increasing/decreasing).
Graph 1. velocity – time
graph for objects rolling
horizontally
Graph 2. acceleration –
time graph for objects
rolling down an incline
34
II.Linear motion down an incline
Release a ball on an inclined board. Draw and label the velocity-time and the
acceleration-time graphs on the axes below.
III. Two-dimensional along an incline
A. Tracing the trajectory
1. Make a marble launcher by attaching the
popsicle sticks to the retractable pen and will serve
as the launching pad of the marble. See Figure 1.
2. On the board, select and draw fixed origins at
points A and B. The left and bottom ends of the
board may serve as the y-axis and x-axis,
respectively.
To complete the setup, elevate one end of the board using books with an angle of inclination
of about 40◦. Get another book to hold the inclined surface, as shown in Figure 3.
3. Push the top end of the improved retractable pen
and firmly hold it horizontally at point A. Then
carefully place the powder-coated marble on its
launching pad. Push the clip of the improved
retractable pen to launch the marble.
4. Trace the path (trajectory) of the marble using a
pencil. Label this path as "horizontally launched" and
set aside later for analysis.
Graph 3. Time graph for
objects rolling straight
down an incline
Graph 4. Acceleration---
time graph for objects
rolling straight down an
incline
Complete the sentence by encircling the answer.
A ball rolling straight down an incline has a velocity that is (increasing,
decreasing) as the object moves (upward, downward), and an acceleration
that is (constant, changing) and (upward, downward).
40º
Figure 2 Set up for projectile motion on an
inclined plane
Source: Science---Grade 9 Learner’s Module
Figure 1Retractable pen attached with
popsicle as launching pad
Source: Science---Grade 9 Learner’s Module
5. Set the powder-coated marble on the launch pad
at point B. Position the launching pad at the
origin. Carefully launch the marble at 15º using
the retractable pen.
6. Trace the path (trajectory) with a pencil. Label
this path as "launched at 15º angle."
7. Do steps 5 and 6 for the other selected angles
(30º, 45 º, 60º, and 75º).
GUIDE QUESTIONS:
(Please attach the graphing paper you used in the activity)
Q1. Describe the path (trajectory) for horizontally-fired projectiles along an incline. Draw the
path (trajectory) of the marble.
Q2. Describe the form of the trajectory for projectiles fired at angles along an incline. Draw
the path (trajectory) of the marble.
Q3. Compare the locations of the trajectory peaks in terms of maximum height reached.
Q4. Compare the horizontal distances (range) reached when they return to the elevation from
which they were launched.
Q5. Look at the path or the trajectories of projectiles fired at angles for the same launching
velocity, which covered the greatest range (horizontal distance in the x-axis)?
Q6. Among the trajectories of projectiles launched at angles, for the same launching speed,
which recorded the highest peak?
Q7. Which pairs of trajectories have almost equal ranges?
DAY TWO ACTIVITY
B. Recording the Hang Time
8. Launch or project the marble at different angles on the inclined board.
Record the hang time (using a stopwatch) of the marble from the time it was released until it
hits the floor.
Supply the table below using the data that you got in this activity.
Safety check
- Ensure that the trajectories are free from hindrances.
Table 1. Hangtime of the marble projected at different angles.
Figure 3 Inclined illustration board supported
between books for the marble projectile
Source: Science---Grade 9 Learner’s Module
Q8. At what angle is the hang time longest? ____º the shortest hang time? _____º
What is it
Projectile Motion: Motion in Two Dimensions
When the player hits the "Sepak,"
you know that it will fall back to the ground
due to the effect of gravity; the motion of the
"Sepak" is called projectile motion. The
"Sepak" itself is called projectile. The
"Sepak" travels in a curved or parabolic path
called trajectory and will also go towards
the ground, especially if air resistance is
negligible. This curved path or parabolic
path has also been observed in Activity 1 as
you tried to launch the marble at different
angles on an inclined surface. Regardless of
the projection angle, the projectile will
always follow a specific path due to the pull
of gravity.
There are two coordinates usually used to describe projectile motion: horizontal and
vertical axes. The horizontal distance traveled by the projectile is called the range. While the
vertical distance, that is, the distance from where it was launched to the topmost point of its
path, is called its height.
Examples of projectiles are cannonball
launched by a cannon, golf ball hit by a
golfer, and an ice skater jumping over
some barrels. Usually, a strong, abrupt
force initiates the motion of a projectile.
Following this force, the projectile moves
through the air and is influenced only by
the earth's gravitational force pulling it
down and by air resistance. If the effect of
air resistance is ignored, equations in free
fall are readily used to analyze the motion
Figure 4 Sepak Takraw Players
Source: https://lrmds.deped.gov.ph/pdf-view/7191
Figure 5 Angle of Projection vs Range
https://lrmds.deped.gov.ph/pdf-view/7191
Figure 6 Matching trajectory A to a half parabola
Source: Science---Grade 9 Learner’s Module
of a projectile – how high it will travel, how far it will go, and so on.
A notable thing to note is that the same range is obtained from two different projection
angles– complementary angles. A body thrown into the air at an angle of 75º, for example, will
have a similar range as if it were thrown at the same speed at an angle of 15º. An object
thrown at 60o will have the same range as when the object is launched at 30o. As you can see,
when we get the sum of 75o angle and 15o angle, 60o angle, and 30o angle, in both sets, we
would obtain a 90o angle. This means that the 75o angle and 15o angle are called
complementary angles. Similarly, 60o angle and 30o angle are also complementary angles.
Thus, complementary angles (angles whose sum is equal to 90o) would result in an equivalent
range. Similarly, you have also observed that the marble, when launched at different angles,
also reached different distances at different times. That is, for smaller angles, the object
remains in the air for a shorter time. A maximum range is attained when an object is launched
45o from the horizontal.
What's More
Activity 2: Curve A Like Objective: The students will be able to set a ball in projectile motion to match pre-drawn
parabolic trajectories.
Materials:
Chalk
manila paper (2 whole sheets)
Small ball or any round object that is safe to throw (e.g., tennis ball, sepak takraw, etc.)
Procedure:
1. Match-a-curve.
a. Draw a rough parabola by drawing
vertical and horizontal lines on manila
paper and throw the ball like in Figure 6.
GUIDE QUESTIONS:
Q1. In what direction or
orientation did you throw the ball?
Q2. How would you describe the
ball's path and motion?
Q3. How many tries did you make
to match the curved paths?
b. Draw a box at the bottom end of the parabola. Throw the ball again with the box as the target.
Q4. How many tries did you make before you matched the curves this time?
13
Figure 7 Matching trajectory B to a half parabola
Source: Science---Grade 9 Learner’s Module
Q5. What does this tell you regarding visuals or imaginary targets in sports?
1. What a curvy-a-throw!
a. On another manila paper, draw a
complete parabola and throw the ball
similar to Figure 7.
Q7. How would you describe the ball's path
and motion?
Q8. Aside from doing more trials or
"practices" for this parabola, where will you
place the imaginary target to aim at for
better matching results?
Q9. Based on the activity, is it possible that the ball will end at a higher elevation than its starting
level?
Q10. What force got the ball projected?
Q11. What force continued to act on the ball when in mid-air?
3. Of curves
a. The drawn curved graphs on the paper are parabolic curves. Similarly, trajectories A and B are also
parabolic curves.
Q12. How will you compare or contrast the horizontal and vertical spacing?
Q13. What does the spacing in the set of vertical lines indicate about the vertical displacement and
vertical velocity of the projectile motion?
4. and arrows.
The displacement,d, and velocity,v, are vector quantities.
Projectile motion can be understood by analyzing the horizontal and the vertical components of the
displacement and velocity, which add as vectors.
Figure 8 Sketch of the velocity-vector component
Source: Science---Grade 9 Learner’s Module
PROJECTILE MOTION CALCULATIONS
Projectile motion can be understood by analyzing the horizontal and the vertical components
of the displacement and velocity, which add as vectors.
A. Projectiles Launched Horizontally (Half- projectiles)
A projectile launched horizontally has no initial vertical velocity. Thus, its vertical motion is
identical to that of a dropped object. The downward velocity increases uniformly due to gravity, and
the horizontal velocity is uniform.
For half-projectiles, you can use the following equations to describe the motion of the projectiles.
Where: h= height at which the projectile is released
ℎ =1
2𝑔𝑡2
g= acceleration due to gravity
t = is the elapsed time
x0 =the horizontal displacement, and;
𝑥𝑜 = 𝑣𝑡 v =horizontal velocity of the projectile.
SAMPLE PROBLEM:
A ball is kicked horizontally off a 20.0-meter high hill and lands a distance of 30.0 meters
from the edge of the hill. Determine the initial horizontal velocity of the ball.
How can the formula be used?
Step 1: Find what is asked in the problem.
You are asked to determine the horizontal velocity of the ball.
Step 2: Identify the given in the problem
h= 20.0 m
x0 = 30.0 m
Step 3: Use the half projectile motion equations to solve for the unknowns.
Before you can find the initial horizontal velocity (v), you must determine first how long (t) the
ball is in mid-air. For the horizontal distance traveled, ℎ =1
2𝑔𝑡2 will be used.
Given: h= 20.0 m
x0 = 30.0 m
Find: v=?
h=1/2gt2
20.0 m= ½ (9.8 m/s2) t2
20.0 m= (4.9 m/s2) t2
t2= 20.0 𝑚
4.9 𝑚/𝑠2
t= √4.0816 s2
t= 2.02 s
Using the time, t=2.02 s, we can now solve the horizontal velocity using the equation, 𝑥𝑜 = 𝑣𝑡
Rearranging the equation, we get, v=𝑥𝑜
𝑡. Plugin the known values, we have:
v=𝑥𝑜
𝑡
v= 30.0 𝑚
2.02 𝑠 = 14.85 m/s
Step 4: Get the answers.
The horizontal velocity of the ball is 14.85 m/s.
B. Projectiles Launched at an Angle
If a projectile is launched upward at an angle, its velocity has two components:
1. A constant horizontal velocity that travels in the same way as the launch,
the acceleration of which is zero; and
2. A rising positive vertical velocity component that is decreasing in magnitude
until it becomes zero at the top of the trajectory (therefore, it no longer goes up any further). But because
gravity makes it accelerates downward at a rate of 9.8 m/s per second or 9.8 m/s2, (therefore it stays at
rest only for an instant), it will start to descend with an increasing negative vertical velocity until it is
stopped by something. So as the projectile moves forward horizontally with uniform velocity, its
vertical velocity is also accelerated, creating a trajectory that is a parabola.
For full projectiles, objects are released at a certain angle from the horizontal. In this case, we can use
the following equations to describe the motion of an object moving in full projectile motion.
Where: θ = launch angle of the projectile,
v0 = initial velocity, and
g =acceleration due to gravity
SAMPLE PROBLEM:
An athlete kicks a ball with an initial velocity of 25 m/s at an angle of 30o with the horizontal. Calculate
the ball's time of flight, horizontal distance, and maximum height.
Step 1: Find what is required in the problem.
Calculate the ball's time of flight, horizontal distance, and maximum height.
Step 2: Identify the given in the problem
v = 25 m/s
θ = 30o
Step 3: Use the half projectile motion equations to solve for the unknowns.
x= 𝑣𝑜 𝑠𝑖𝑛2𝜃
𝑔
x = (25 m/s)2 sin (2(30))
(9.8 m/s2
x= (25 m/s)2 sin (60)
(9.8 m/s2
x= 55. 23 m
Step 4: Get the answers.
The time of flight is 2.55 s, the maximum height is 7.97 m, and the horizontal range is 55.23 m.
What I Have Learned
For numbers 1-4, complete the sentence by encircling the best answer.
1. A (projectile, parabola) is an object upon which the only force is gravity.
2. Projectiles travel with a (parabolic, straight) trajectory due to the influence of gravity.
3. When the initial launching angle is greater, the (shorter, longer) the range will be.
4. If a ball is thrown at a 15-degree angle, it will have a (shorter, longer) range and
height than a ball thrown at a 45-degree angle.
5. What do you think?Two balls are set to move off a table. One is released, while the
other is given an initial horizontal velocity. Which ball will reach the ground first?
Explain.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
What I Can Do
Projectile motion is a beneficial and practical concept in Physics. For example, if there
are floods and rescuers could not reach the place, a rescue plane is usually used to drop a
package of emergency rations to the victim. Can you think of some other ways how the concept
of projectile motion is helpful in real-life situations?
5
5
Assessment
I. Choose the best answer in each item by writing the letter before the number.
1. At what degree should a water hose be pointed in order for the water to land with the
greatest horizontal range?
a. 0° c. 30°
b. 45° d. 60°
2. Given the same initial velocity, at what another angle
should a ball be hit to reach the same distance if it is being
shot at an angle of 30o and it reaches a distance of 50 m.
a. 15° c. 45°
b. 60° d. 75°
3. When objects are undergoing projectile motion, what
do you call the force acting on them?
a. Air Drag
b. Normal Force
c. Air Resistance
d. Gravitational Force
4. What do you call objects moving in two dimensions?
a. Trajectory
b. Free-body
c. Projectile
d. Parabola
5. What do you call the path taken by an object moving in projectile motion?
a. Gravity b. Projectile c. Trajectory d. Force
6. In a place where gravity doesn't exist, what will happen to a ball thrown upward?
a. It will continuously move upward. c. It will follow a parabolic path.
b.It will fall down. d. It will float.
7. What is the value of acceleration due to gravity equal to?
a. 0 m/s2 b. 9.8 m/s2 c. 9.8 m d. 9.8 m/s
8. What is the acceleration of a sepaktakraw ball that is hit vertically upward by a player
after 1 second?
a. 0 s b. 1 m/s2 c. 9.8 m/s2 d. 9.8m
9. At which part of the path does a projectile have a minimum speed?
a. When it is thrown
b. Half-way to the top
c. At the top of its path
d. When it returns to the ground
10. A projectile is thrown 30º above the horizontal. What happens to its acceleration as it
moves upward?
a. It decreases because its velocity is directed upward
b. It increases because its velocity is directed upward
c. It decreases because its velocity is decreasing
d. It remains the same
20
15
Source: Science---Grade 9 Learner's
Module
II. Solve the following problems
For number 1-3: A football player kicks a football from the ground level with an initial
velocity of 27 m/s, 30 degrees above the horizontal.
1. What is the maximum height (h) the ball attained?
a. 2.44 m b. 7.89 m c. 9.30 m d. 20.80
2. How many seconds did it take the football to return to the launching height?
a. 76 s b. 1.76 s c. 2.76 s d. 3.76 s
3. How far away did it land (X)?
a. 64.42 m b. 75.0 m c. 100.0 m d. 42.44 m
For numbers 4-5: A physics book slides off a horizontal tabletop with a speed of 1.10
m/s. It strikes the floor in 0.350 s.
4. What is the height of the table above the floor?
a. 0.25 m b. 0.44 m c. 0.50 m d. 0.60 m
5. What is the horizontal distance from the edge of the table to the point where the book
strikes the floor?
a. 0.30 m b. 0.39 m c. 1.0 m d. 1.5 m
Additional Activities
A bowling ball unintentionally falls out of an
airliner's cargo bay as it flies along in a horizontal
direction. As detected by a person standing on the
ground and viewing the plane as in the figure at right,
which path would the bowling ball most closely
follow after leaving the airplane. ___________
5
Answer Key- Gr9Q4W2 Science
What’s New Activity 1: Curve Me on an
Incline
I.Constant, zero
II.Increasing,
downward;
constant, downward
Activity 1: Curve Me on an
Incline
Guide Questions:
Q1: The trajectory is a half open-
down parabola. Other students
may answer curve down or
concave down.
Q2. All the trajectories are full
open-down parabolas. In
addition, some students may also
state something about different
maximum heights, etc.
Q3. The trajectory peaks for each
projection angle do not have the
same location. The peaks are
closest to the y-axis origin for
shortest range or greatest angle
of projection. Each peak is
reached just before half the range
was travelled. This indicates
frictional forces between marble
projectile and inclined surface
resulting to a not so perfect open-
down parabola.
Q4. The trajectories have different
horizontal distances (range) reached,
but some ranges are quite short,
some extend beyond the board or
cookie sheet.
Q5. The trajectory fired closest to or
at 450 covered the greatest range.
Q6. The trajectory with the greatest
launching angle recorded the highest
peak.
Q7. Trajectories at 150and 750 have
almost similar ranges. Trajectories at
300 and 600 also have almost similar
but longer ranges than those for 150
and 750. Some students may note
close ranges for pairs of angles that
are almost if not complementary
angles.
Q8. The average range is longest
for the highest drop at 2 m and
shortest at a 0.5 m height of fall.
What’s More
Activity 2: Curve Me on an Incline
Q1. The ball was thrown
horizontally from the top
Q2. The ball’s path is curved
downwards similar to the drawn
graph. At the start, it moved
horizontally forward but as it moved
forward, it also moved downward.
Q3. (Depends on the thrower’s
skills.)
Q4. (Depends on the thrower’s
skills, but predictably lesser tries
than before because of the visual
goal.)
Q5. Aiming at visual goals makes
practice easier and results in better
approximations of flight.
Q6. The ball was thrown upward
from the bottom left at an angle from
horizontal.
Q7. The ball moved up in a curved
path until it reached a maximum
height and then it moved downward
still following the curved path.
Q8. It is best to have an
imaginary target at the top of the
curve rather than anywhere else
along the parabola.
Q9. In both throws the balls
always end up on a lower
elevation. It is not possible that
the ball will end at a higher
elevation than its starting level.
Q10. The initial push from the
throw.
Q11. The force of gravity acted
at all times on the ball.
Q12. The spacing between
horizontal lines is equal unlike
the spacing between vertical
lines which increases by the
square of a span/unit.
Q13. The increasing distance
between vertical lines indicate
that the vertical motion is
accelerated due to gravity.
What I Have Learned
1.Projectile
2.Parabolic
3.Shorter
4.Shorter
5.Both balls will reach the
ground at the same time
since both balls are acted
by the same gravitational
force
What’s I Can Do Answers may vary.
Assessment
I. II.
1.B 1. C
2.B 2. C
3.D 3. A
4.C 4. D
5.C 5. B
6.A
7.B
8.C
9.C
10.D
References Books:
Science Learners Material Grade 9, Pages 242-255
Science Teacher’s Guide Grade 9, Pages 168-184
Electronic Resources:
“LR Portal.” Deped LR Portal. Department of Education. Accessed December 28,
2020. https://lrmds.deped.gov.ph/detail/7191.
Development Team Region IX Hymn
Writer:
Editors: IRMINA C. CALIBO
Reviewer: Mila P. Arao, EPS Illustrator: Layout Artist: Management Team:
Danny B. Cordova, EdD,CESO VI SDS
Ma. Coleen L. Emoricha, EdD, CESE ASDS
Ma. Diosa Z. Peralta CID Chief
Ma. Madelene P. Mituda, EdD EPS-LRMDS
Mila P. Arao EPS -Science
OUR EDEN LAND
Here the trees and Golden beams of
flowers bloom, sunrise and sunset, Here the breezes Are visions you'll never gently blow, forget.
Here the birds sing Oh! That's Region IX...
merrily, And liberty forever Hardworking people stays, abound,
Every valley and dale
Here the Badjaos Zamboangenos, swam the seas, Tagalogs, Bicolanos,
Here the Samals live in Cebuanos, Ilocanos,
peace, Subanens, Boholanos, Here the Tausogs Illongos, thrive so free, All of them are proud With the Yakans in and true
unity. Region IX our Eden Land.
Gallant men And Ladies fair, Linger with love and Region IX, our Eden
care, Land.
Teacher III Zamboanga del Sur National High School
RUBIE MAE C. RESTAURO Teacher III Co Tek Chun National Trade School
