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First Quarter – Module 4: Week 4
Motion in Two and Three Dimensions
NOT FOR SALE
A Joint Project of the
and the
GENERAL PHYSICS 1 - Grade 12
Alternative Delivery Mode
Quarter 1 – Module 4: Motion in Two and Three Dimensions
First Edition, 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of
the Government of the Philippines. However, prior approval of the government agency or office
wherein the work is created shall be necessary for exploitation of such work for profit. Such
agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,
trademarks, etc.) included in this module are owned by their respective copyright holders. Every
effort has been exerted to locate and seek permission to use these materials from their
respective copyright owners. The publisher and authors do not represent nor claim ownership
over them.
Published by the Department of Education
Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio
Printed in the Philippines by ________________________
Department of Education – Region IX
Office Address: ____________________________________________
Telefax: ____________________________________________
E-mail Address: ____________________________________________
Development Team of the Module
Writers: Jeovanny A. Marticion
Editors:
Reviewers: Cyrus A. Ratilla
Illustrator:
Layout Artist:
Management Team: Virgilio P. Batan - Schools Division Superintendent
Jay S. Montealto - Asst. Schools Division Superintendent
Amelinda D. Montero - Chief Education Supervisor, CID
Nur N. Hussien - Chief Education Supervisor, SGOD
Ronillo S. Yarag - Education Program Supervisor, LRMS
Leo Martinno O. Alejo - Project Development Officer II, LRMS
What I Know
Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1.Which of the following is TRUE when a projectile is launched at an angle above the
horizontal and reaches its maximum height?
a.accelerations in x and y components are zero
b.y-acceleration is 9.8 m/s
2
and zero along x
c.x-acceleration is 9.8 m/s
2
and zero along y
d.both x and y-acceleration is 9.8 m/s
2
2.A player threw two balls with the same force wherein one thrown at 45° and the
other at 60°. Between the two balls, the one that will have a longer horizontal range is
a.45°
b.60°
c.both
d.either 45° or 60°
3.The horizontal velocity component of a projectile (ignoring air resistance__.
a.remains the same
b.continously increases
c.zero
d.decreases
4.A bullet is fired horizontally and at the same instant velocity a second bullet is
dropped from the same height. Ignoring air resistance, which is true?
a.fired bullet hits first
b.they hit the same time
c.dropped bullet hit first
d.incomplete info
5.The acceleration due to gravioty in the Moon is only 1/
th
of the Earth. If you hit a
baseball with the same effort that you would on Earth, the ball would land
a.one-sixth as far
b.same distance
c.6 times as far
d.36 times as far.
6.The angle for a ball to be thrown to reach the maximum horizontal distance is____.
a.0°
b.30°
c.45°
d.90°
7.This refers to an object launched and follows a curved path while influenced by
gravity.
a.object
b.particles
c.projectile
d.free falling bodies
For nos. 8-10, refer to the problem below:
A shell is fired at a velocity of 300 m/s at an angle of 30° above the horizontal.
8 .How far does it go?
a.7 953 m
b.4 591 m
c.26.51 m
d.4 592 m
9 .What is its time of flight?
a.61 s
b.31 s
c.15 s
d.25 s
10 .What is its maximum altitude?
a.1 148 m
b.7.65 m
c.15.31 m
d.2 296 m
11.Which of the following statements is FALSE about the cannon ball’s path thrown
horizontally.
a.it has a uniform velocity along the x component
b.velocity along the y component increases with respect to time
c.velocity along the y component decreases with respect to time
d.the range depends on its initial velocity
12.The ball was released off the table. How much time it takes to reach the ground?
a.
ℎ
𝑣
b.
1
2
ℎ
𝑣
c.√
2ℎ
𝑔
d.√
𝑔
2ℎ
13.A ball is tied on a string and swung in a vertical circular motion. When it reaches
the peak, its acceleration vector is represented by:
a. b. c. d.
14.A ball rotates at a speed of 3 m/s tied on 1.2 m string. What is the centripetal
acceleration of the object? a.1.2 m/s
2
b.3.0 m/s
2
c.7.5 m/s
2
d.3.2 m/s
2
15.A girl whirls the ball at the end of a string. Which of the following statement is
a.speed is not constant
b.velocity is not constant
c.radius is constant
d.acceleration varies
displacements relative to the flatcars and their (flatcars and motorcycle) displacements
relative to the Earth. Hence,
𝑋
𝑚𝑜𝑡𝑜𝑟𝑐𝑦𝑐𝑙𝑒/𝐸𝑎𝑟𝑡ℎ
= 𝑋
𝑚𝑜𝑡𝑜𝑟𝑐𝑦𝑐𝑙𝑒/𝑓𝑙𝑎𝑡 𝑐𝑎𝑟𝑠
𝑓𝑙𝑎𝑡 𝑐𝑎𝑟𝑠/𝐸𝑎𝑟𝑡ℎ
𝑑 𝑥
𝑚/𝐸
𝑑𝑡
𝑑 𝑋
𝑚/𝑓
𝑑𝑡
𝑑 𝑋
𝑓/𝐸
𝑑𝑡
, thus:
𝒗
𝒎/𝑬
= 𝒗
𝒎/𝒇
𝒇/𝑬
If the motorist is moving at 13 m/s and the train at 30 m/s to the right, then
the velocity m/E (motorist with respect to Earth) is moving at 43 m/s to the right. If
the motorist is moving at same velocity but this time to the left, the velocity m/E
(motorist with respect to Earth) is moving at 17 m/s to the left. It is just the algebraic
sum of the velocities.
If you notice we used subscripts on their velocities. For example, m/E means
velocity of motorcycle with respect to Earth. In writing this equations, make sure that
the first subscript on the left side of the equation is the first subscript on the first term
of the equation while the second subscript at the right side of the equation is the
found at the right side of the second term in the equation.
Problem You drive north on a straight-two lane road at a constant 88 km/h. A
truck in other lane approaches you at a constant 104 km/h.
What is
asked?
Find the truck’s velocity relative to you. 𝑣
𝑡𝑟𝑢𝑐𝑘/𝑦𝑜𝑢
What is
given?
𝑡𝑟𝑢𝑐𝑘/𝑒𝑎𝑟𝑡ℎ
𝑦𝑜𝑢/𝐸𝑎𝑟𝑡ℎ
Diagram
Strategy There are three perspectives in the given problem: truck, you and Earth.
Solution 𝑣 𝑡𝐸
𝑡𝑦
𝑦𝐸
𝑡𝑦
𝑡𝐸
𝑦𝐸
Answer Therefore, the truck’s velocity with respect to your perspective is
moving at - 192 km/h
What is
asked?
Your velocity relative to truck
What is
given?
Strategy Your velocity can always be the same with your perspective to any object
except that you and the truck were moving in opposite directions.
Solution
𝑡/𝑦
𝑡/𝑦
v = 30 m/s
ImageSource:https://www.google.com/search?q=flatcars+and+their+(flatcars+and+motorcycle)+displacements+relative+to+the+Earth&tbm=isch&ved=2ahUKEwigicTa4bjrAhUL5JQKHQyPAN4Q2-
cCegQIABAA&oq=flatcars+and+their+(flatcars+and+motorcycle)+displacements+relative+to+the+Earth&gs_lcp=CgNpbWcQA1Cu9gNYrvYDYIv8A2gAcAB4AIABAIgBAJIBAJgBAKABAaoBC2d3cy13aXotaW1nwAEB&sclient=img&ei=KEd
GX6DVO4vI0wSMnoLwDQ&bih=527&biw=616#imgrc=DhgTCnZKrn7EJM
What’s More
Solve the problem below. Write your answers on the provided answer sheet.
An airplane’s compass indicates that the is headed due north and is airspeed indicator
shows that it is moving through the air at 240 km/h. If there’s a 100 km/h wind from
west to east, what is velocity of the airplane relative to earth?
What I Have Learned
Fill in the blanks. Write your answers on the provided answer sheet.
Suppose a woman is driving a car. She was spotted by a patrol officer when she
was driving beyond the road’s limit. Prior to pursuit, the woman has a velocity with
respect to ______________________. On the other hand, motorist has a velocity with
respect to ___________________ and ____________________. ___________________ velocity
refers to velocity of one’s body relative to other.
What I Can Do
The idea of relative motion has been considered as today’s complicated concept
in Physics. There are 3 main problems here: how the event took place, how observers
in relative motion see it? And when do they see this event? Provide a situation where
these questions could be applied. Write your answers on the provided answer sheet.
range
y
x
x
y
x
y
x
y
x
reasonable mass moves with lower speed. On the other hand, objects with higher
speed and forces becomes a significant factor, the idealized model for projectile model
could not anymore fit. This is also true for lighter objects due to influence of external
forces which could affect the movement of the ball. Therefore, idealized model
considers weight as the only force.
The projectile motion is a combination of horizontal and vertical components of
motions with constant acceleration. These are independent from each other and we
will analyze them separately. These motions are just superimposed from each other.
The figure below shows the direction of velocity components along x and y-axis.
Acceleration vectors can then be shown
using the figure below. Since velocity
along x-component is constant, the
acceleration is zero. On the other hand,
velocity along y-component is moving at
constant acceleration. The value of
acceleration is equal to 9.8 m/s
2
.
Along x – axis
𝒙
𝟎𝒙
𝒐
𝟎𝒙
Along y – axis; where a = - g
𝒚
𝟎𝒚
𝟎
𝟎𝒚
𝟏
𝟐
𝟐
range
y
x
x
y
x
y
x
y
x
The previous case is the simplest example for a projectile motion. This is when you
usually release a ball in a trajectory path or a bullet fired from a horizontal gun and
leaves it moving in trajectory path. The figure below shows another case of projectile
motion thrown at some angle. The projectile trajectory is a parabola. The velocity along
y component decreases and becomes zero when it reaches the highest peak. When it
goes back to Earth’s ground, the velocity along y component increases but directed in
opposite direction. The velocity along x component, on the other hand, is constant
throughout the path.
From equation 2,
we substitute vox
𝑜
0 𝑥
Where, 𝑣
0 𝑥
0
𝒐
𝟎
From equation 4,
we substitute v oy
0
0 𝑦
2
Where 𝑣
0 𝑦
0
𝟎
𝟎
𝟏
𝟐
𝟐
distance r of the projectile from
the
𝟐
𝟐
The speed of a projectile at any
given time
𝒙
𝟐
𝒚
𝟐
The direction of projectile in terms
of angle
𝒗
𝒚
𝒗
𝒙
Along x - axis
𝑥
0 𝑥
0 𝑥
0
Then, 𝒗
𝒙
𝟎
𝑥
𝑦
y
x
y
x
y
x
x
y
x
y
x
maximum height
Along y-axis
𝑦
0 𝑦
Where 𝑣
0 𝑦
0
Then, 𝒗
𝒚
𝟎
Similarly, in the
vector resolution
lesson, you need to
analyze the vector
using right triangles.
Ɵ
For a projectile launched with initial velocity v 0 with an angle Ɵ from the horizontal,
we can derive general expression for the maximum height.
𝑓
2
𝑜
2
Since we are dealing with motion along the y-axis, then initial velocity along y is given
by 𝑣 𝑜
𝑠𝑖𝑛Ɵ , the acceleration is equal to g and displacement is represented by h,
maximum height.
𝑓
2
𝑜
2
The velocity along y axis at maximum height is zero while x component is still
constant. Then,
𝑜
2
𝑜
2
𝑜
2
Derive general expression for the maximum horizontal range R
𝑜
0
𝑜
0
𝑜
0
0
0
2
0
0
0
0
2
0
0
0
2
0
0
2
0
2
2
0
0
2
0
2
2
0
0
0
2
2
0
2
2
0
2
2
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
1
2
𝑅
𝑣
0
2
𝑐𝑜𝑠
2
𝜃
0
2
2
0
2
2
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
1
2
𝑅
𝑣
0
2
𝑐𝑜𝑠
2
𝜃
0
2
2
0
2
0
2
0
2
0
2
Range along x-
axis
solve for t
Range along y-
axis
substitute
𝑅
𝑣
0
𝑐𝑜𝑠𝜃
eliminate “R”
eliminate “𝑣
0
" on the
left side of the
equation
solve for R
since
𝑠𝑖𝑛 2 𝜃 = 2 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃
Problem A motorcycle stunt rider rides off the edge of a cliff. Just as the edge
of its velocity is horizontal with magnitude of 9.0 m/s. Find the
position, distance from the edge of cliff and
of 0.5 s after it leaves the edge of the cliff.
What is
asked?
x =?, y =? at 0.5 s,
What is
given?
initial velocity of 9.0 m/s
Strategy?
𝑜
0 𝑥
0
0 𝑦
1
2
2
Solution
𝑜
0 𝑥
0
0 𝑦
1
2
2
1
2
2
1
2
𝑚
𝑠
2
Answer The stunt rider is located (4.5 m, - 1.225 m) at t = 0.5s
What is
asked?
velocity at t = 0.5 s
What is given? initial velocity of 9.0 m/s
Strategy?
use equations 𝑦 = 𝑦
0
0
1
2
𝑔
2
and 𝑣 = 𝑣
0
𝑔
Solution solving for x component
of velocity
𝑥
0 𝑥
solving for y component
of velocity
𝑦
0 𝑦
2
solving for magnitude
𝑥
2
𝑦
2
2
2
solving for direction
∝ = tan
− 1
𝑦
𝑥
= tan
− 1
Answer The stunt rider has a velocity of 10.24 m/s before it reachers
the ground directed at 26°
Problem A batter hits a baseball so that it leaves the bat at a speed of 𝑣 0
= 37. 0 𝑚/𝑠
at an angle of 53.1°.
What is
asked?
Find the position of the ball and its velocity at t = 2 s
What is given? 𝑣
0
= 37. 0
𝑚
𝑠
Ɵ = 53.1°
Solution we use eq’n 5 to
solve x
component
𝑣
𝑥
= 𝑣
0
𝑐𝑜𝑠𝜃 = ( 37
𝑚
𝑠
) (cos 53 .1°) = 22. 2 𝑚/𝑠
we plug-in the
value of v x
in
eq’n 7
𝑥 = 𝑥
𝑜
0 𝑥
𝑡 = (
𝑠
) ( 2 𝑠) = 44. 4 𝑚
we use eq’n 6 to
solve for
position at y
𝑣
0
𝑠𝑖𝑛𝜃 = ( 37
𝑚
𝑠
) (sin 53 .1°) = 29. 6 𝑚/𝑠
𝑦 = 𝑦
0
0 𝑦
𝑡 −
1
2
𝑔𝑡
2
= ( 29. 6
𝑚
𝑠
) ( 2 𝑠)
2
= 39. 6 𝑚
ImageSource:https://www.stevespanglerscience.com/la
b/experiments/the-coin-drop-sick-science/
What I Have Learned
Fill in the blanks. Write your answers on the provided answer sheet.
An object traveling in a curved path is called ___________________. The motion is
called _________________________. The path it travels is called _____________________.
When an object is thrown and follows this motion, the vertical component of its
acceleration is equal to ___________ while the horizontal component of its acceleration
is equal to ___________________. This is due to the fact that horizontal component of its
velocity is always ____________ while its vertical component is ____________. The vertical
component of velocity ____________ when it goes upward. It becomes _________ when it
reaches the peak. Upon returning to the ground, the velocity _________________ and
directed __________________.
What I Can Do
You will be observing two coins dropping from a table. Coin A will be dropped from the
table while the other one will be projected from the table. You can use a phone camera
to take a video of the demonstration
floor. Coin A must be dropped directly while
Coin B must be thrown in a projectile. You can
do this by flicking the coin across the table to
strike the first coin at the table’s edge. Try to
aim “off center”, this will drop the coin straight
down while projecting the coin with some
horizontals peed.
drop the floor. You may ask assistance of two
person in your house or someone to record the time.
Record your data on the provided answer sheet.
Calculate the time it will take for the coin to drop using equations for free-falling
bodies and projectile motion. Assume a zero initial velocity for both coins.
Compute for the percentage difference between calculated time and the experiments.
Show your solutions on the provided answer sheet.
Lesson
3
Circular Motion
What’s In
The previous module discussed how the components of object’s motion changes
when it undergoes projectile motion. The motion is uniformly accelerated throughout
its journey due to influence of Earth’s gravity. This lesson will discuss circular motion
which is also influenced under constant acceleration although it moves at constant
speed.
What’s New
Satellites are considered as projectiles. These satellites are objects where the
only force experienced is the Earth’s gravity. When a projectile is launched with a
enough speed, it will orbit around the Earth. If the launch speed is too small, this will
lead the satellite to fall towards Earth. Hence, speed must be calculated carefully to
ensure it would not fall back toward Earth and will just maintain its height
throughout its journey.
What is It
When a particle moves in a circular path at
constant speed, the motion of the object is said to
undergo uniform circular motion. The acceleration
is not parallel to the path. It is always directed
towards the center. We will derive the relation.
Acceleration is always perpendicular to the velocity
vector and it changes direction continuously.
a
a
ImageSource:https://www.google.com/search?q=circular+motion&rlz=1C
1JZAP_enPH705PH706&hl=fil&sxsrf=ALeKk02NOLCJhRgLmVaS0Q5rCe
5WbBJLw:1598442875215&source=lnms&tbm=isch&sa=X&ved=2ahUKE
wiQjJXe57jrAhUxGqYKHRIOD4UQ_AUoAXoECA0QAw&biw=616&bih=
7#imgrc=jmZZGPp4vrOStM
What’s More
Solve the given problem. Write your answers on the provided answer sheet.
A Ferris wheel with radius 15.0 m is turning about a horizontal axis through its
center; the linear speed of the passenger rim is constant and equal to 9 m/s.
What are the magnitude and direction of the acceleration of a passenger as she passes
through the lowest point in her circular motion?
How much time does it take the Ferris wheel to make one revolution?
What I Have Learned
Fill in the blanks. Write your answer on the provided answer sheet.
An object undergoing uniform circular motion has a acceleration pointed at the
_____________ although the speed is ___________________. The time it takes to complete
one full circle is called _________________________.
What I Can Do
A viral video of a Physics professor in Silliman University is currently circulating
because of his demonstration on circular motion. He securely tied the handle of a pail
through a string. Then, he filled it with water. He reminded the students that he will
swirl the system of objects and assured they will not get wet. Everyone was screaming
and when he was finished, the class was shocked no amount of water was spilled
throughout the demonstration. Why do you think this happened? Write your answer
on the provided answer sheet.
Assessment
1.What is the projectile’s horizontal accelerations when it was thrown at an angle of 30
degrees above its horizontal?
a.zero b.9.81 m/s
2
c.it varies d.insufficient information
2.A player kicks the ball with a velocity of 25 m/s directed 53 degrees above the
horizontal. What is the vertical component of its initial velocity?
a.15 m/s b.20 m/s c.33 m/s d.25 m/s
3.At what other angle will the football be kicked to travel 50 yards if its initial velocity
was the same with the ball kicked at 25 degree and travels 50 yards?
a.90 degrees b.45 degrees c.55 degrees d.65 degrees
4.Two balls were thrown horizontally from the same height. Ball A has speed of 0.
m/s while ball B has a speed of 20 m/s. The time takes for Ball B to reach the ground
compared to Ball A is
a.same b.twice c.half d.four times
5.The ball was fired initially at 12 m/s from a cannon facing northwards. The cannon
moves eastward at 24 m/s. Which of the vectors represent the resultant velocity of the
ball?
a. b. c. d.
6.An arrow was thrown at angle of 45 degrees while the other arrow was thrown at 60
degrees. Compared to arrows fired at 60 degrees, the arrow fired at 45 degrees is
a.longer time of flight and range c.longer time of flight and shorter range
b.shorter time of flight and longer range d.shorter time of flight and range
7.A ball is thrown at 38 degrees. What happens to its velocity?
a.it decreases then increases c.it decreases then remains the same
b.it increases then decreases d.it increases the remains the same
demonstration of motion. When we release the
lever, the rod releases ball B while the ball A will
be thrown horizontally. Which of the following
statements is true?
a.Ball A travels with constant velocity
b.Ball A and Ball will hit the table top at the
same time
c.Ball B will touch the table top before Ball A
d.Ball B will have an increasing acceleration
ImageSource:https://www.google.com/search?q=compressed+spring+lever+suppot+rod+support+base+&tbm=isch&ved=
ahUKEwi149Xu4rjrAhUHvJQKHbMZDKMQ2-
cCegQIABAA&oq=compressed+spring+lever+suppot+rod+support+base+&gs_lcp=CgNpbWcQA1Dx-
iRYrtElYNfUJWgAcAB4AIABAIgBAJIBAJgBAKABAaoBC2d3cy13aXotaW1nwAEB&sclient=img&ei=X0hGX_W3J4f40gSzs7C
YCg&bih=527&biw=616&rlz=1C1JZAP_enPH705PH706#imgrc=mzF5oVrWuKoB7M
