Republic of the Philippines
Mountain Province State Polytechnic College
Bontoc, Mountain Province
KINEMATICS OF DYNAMICS
Module 1 of 8 Modules
General Physics
Elsa B. Daguio
Messenger account: Herzelle Bag-ay Daguio
Engineering Department
First Semester, School Year 2022-2023
INTRODUCTION
Ask most people what they know about physics, and you‟ll probably get an
answer like, “it‟s hard with lots of math.” That answer, although it has some truth to
it, misses the big picture. Physics is the science that uses observation and reasoning
to explain why things happen in the real world and how to predict what will happen
next.
In this course you will be studying “mechanics”, which explains how and why
things move. Mechanics explains why a quarterback who throws a tight spiral has
better accuracy than a quarterback who can‟t. It explains why the top rung of a ladder
has a warning sticker not to sit or stand there. Mechanics is how your insurance
company knows that you were going at least 90 mph when your car hit that tree, even
though nobody was there to see it but you and you swear you were only doing 45 mph.
In this module we will be emphasizing how to use reason and similarities to
make tough problems simpler before we even do any math. I don‟t expect you to
memorize a load of equations, but I will require you to use your noggin.
This first module will tap into your prior knowledge by linking concepts you
have already studied to some new ideas and applications. It will expand your
knowledge, revisit some ideas already considered, as well as introduce new topics.
The lessons here are measurement, vectors and scalars, linear motion, free fall
and projectile motion. The activities here are designed for all students, regardless of
gender and cultural background. The number of hours allotted for this module shall
be 30 hours and you are expected to finish this on or before October 8, 2021.
LEARNING OUTCOMES
At the end of the module, you should be able to:
1.
calculate the components of vectors applying the basic trigonometric
functions;
2.
calculate the resultant of a vector applying the Pythagorean Theorem;
3.
calculate the displacement, velocity and acceleration of an object in linear
motion using the given formulas;
4.
calculate the displacement, velocity and time of a free-falling body thrown
vertically upward or downward using the given formulas;
5.
calculate the displacement, velocity and time of a projectile thrown at an
angle θ with the horizontal using the given formulas; and
PRETEST
Let see if you are knowledgeable with measurements, vectors and scalar, linear
motion, free-falling bodies and projectile. Take time to answer the following questions
by encircling the letter of your answer before proceeding to the given lessons.
1.
Which of the following is the equivalent unit for mass?
a. Kilogram
b. Liter
c. Meter
1
2.
Which of the following is a direction?
a. North
b. 60°
c. All of the above
3.
Which of the following is used to represent a direction?
a. Arrow
b. Straight Line
c. Point
4.
Which of the following causes an object that is thrown upward to fall back to
Earth?
a. Gravity
b. Mass
c. Volume
5.
Which of the following causes any object to move?
a. Acceleration
b. Force
c. Speed
6.
When you throw a stone at a certain angle then fall back to Earth, how will its
path look like?
a. Circular
b. Parabolic
c. Straight
7.
In
a.
b.
c.
8.
Which of the following is NOT a measurement of Length?
a. Area
b. Displacement
c. Distance
9.
Which of the following measurements has a direction?
a. Distance
b. Mass
c. Weight
10.
Which of the following does not change when an object is set in motion?
a. Distance
b. Mass
c. Velocity
terms of motion of an object, which of the following does “how long” refer to?
Distance
Time
Velocity
2
LESSON 1: MEASUREMENT
Objectives:
At the end of the lesson, you should be able to:
1.
distinguish one measurement from the other; and
2.
convert one unit to another.
LET’S ENGAGE
What are the different measurements in physics? See the following measurements
below. Can you tell what measurements are these?
 5 ounces
 4 hours
 3 yards
LET’S TALK ABOUT IT
In physics, the three basic measurements we must bear in mind are the length,
mass, and time. These measurements are widely used all throughout this course.
Length is defined as the straight – line distance between two points along an object.
Length could be a distance or a displacement, a width and a height. So for example in
a problem, if the question says “how far” or “how high”, it means that it is asking
about length. The following tables show the conversion of units used for Length
measurements.
Table 1.1: Units for length
International System
Metric System
Meter
Centimeter
English or Standard System
Foot
Table 1.2: Metric Conversions
( )
(
( )
)
(
( )
)
( )
(
)
( )
(
)
( )
)
( )
(
(
)
(
(
)
)
(
(
( )
( )
)
)
( )
( )
( )
Table 1.3: English Conversions
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
3
Table1.4: Metric to English Conversions
(
)
( )
( )
( )
( )
( )
( )
( )
Table 1.5: English to Metric Conversions
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
Table 1.6: Metric Conversions for Areas
(
)
( )
( )
(
)
(
(
)
(
)
(
)
)
Table 1.7: English Conversions for Areas
( )
(
(
)
( )
(
)
(
(
)
(
(
)
)
)
)
Table 1.8: Metric to English Conversions for Volumes
( )
( )
( )
( )
( )
( )
( )
( )
Mass is the quantity of matter in a body regardless of its volume or of any
forces acting on it. The following tables show the conversion of units used for Mass
measurements.
Table 1.9: Units for Mass
International System
Metric System
kilogram
gram
English System
pounds
Table 1.10: Metric Conversions in grams
( )
( )
(
)
(
( )
)
( )
)
( )
( )
( )
(
(
)
( )
(
)
(
)
( )
( )
4
(
)
(
(
( )
)
( )
)
( )
Table 1.11: English Conversions
( )
( )
(
)
( )
( )
( )
( )
( )
(
)
( )
Table 1.12: Metric to English Conversions
( )
( )
( )
( )
(
)
(
)
Table 1.13: English to Metric Conversions
( )
( )
( )
( )
(
)
( )
( )
(
)
Time is a measurable period during which an action or something occurs. So
for example in a problem, if the question says “how long”, it means that it is asking
about time. The following tables show the conversion of units used for Time
measurements.
Table 1.13: Units for Time
International System
Metric System
second
second
Table 1.14: Time Conversions
(
)
( )
( )
( )
( )
( )
English System
second
( )
(
)
( )
( )
(
)
( )
(
)
Examples:
1.
How many micrometers are there in
?
 From Table 1.2, the unit conversions to be used will be the following:
(
)
( )
( )
( )
Applying the unit conversions:
(
)(
)
(
(
)(
)
)
5
2.
How many kilometers are there in
?
 From Tables 1.3 and 1.4, the unit conversions to be used will be the following:
( )
( )
( )
( )
Applying the unit conversions:
(
)(
(
)
)(
)
(
)
Alternative solution:
 From Tables 1.2 and 1.4, the unit conversions to be used will be the following:
( )
( )
( )
( )
(
3.
)(
)(
)(
) (
(
)
)(
(
)(
)(
)
)
(
)
)
)(
(
) (
)
How many cubic meters are there in
cubic centimeters?
 From the Table 1.2, the unit conversion to be used will be the following:
( )
( )
(
6.
)
)
How many acres are there in
square meters?
 From Tables 1.5 and 1.7, the unit conversions to be used will be the following:
(
)
( )
Applying the unit conversions:
(
5.
)(
(
How many square millimeters are there in 1.8 hectares?
 From Table 1.6, the unit conversions to be used will be the following:
( )
( )
( )
(
)
(
)
(
)
Applying the unit conversions:
(
4.
(
)
(
)
)
How many kilograms are there in
?
 From Tables 1.11 and 1.12, the unit conversions to be used will be the
following:
( )
( )
( )
( )
( )
( )
(
)(
)(
)
(
(
)(
)(
)(
)
)
6
7.
How many days are there in
?
 From the Table 1.14, the unit conversions to be used will be the following:
Applying the unit conversions:
(
8.
)(
(
)
)(
(
)
)
(
) in meters per second ( ⁄ ).
Convert
 From Tables 1.2 and 1.14, the unit conversions to be used will be the
following:
( )
( )
( )
( )
Applying the unit conversions:
(
)(
(
)
)(
)(
(
)
)
⁄
9.
Add
to
.
 From Table 1.13, the unit conversion to be used will be the following:
Applying the unit conversion:
(
)
Then add the result to 2 grams.
IT’S YOUR TURN
Exercise 1.1. Convert the following.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Exercise 1.2. Convert the following.
( ⁄
11.
(
)
12.
( ⁄ )
13.
( ⁄
)
)
( ⁄ )
(
)
7
LESSON 2: SCALARS AND VECTORS
Objectives:
At the end of the lesson, you should be able to:
1.
perform addition of vectors;
2.
resolve a vector into components;
3.
calculate the magnitude and direction of the resultant vector; and
LET’S ENGAGE
Can you tell which of the following quantities are scalars and which are vectors?
 4 meters
 6 meters/second to the North
 10 N at 30 with the horizontal
 100 Joules
LET’S TALK ABOUT IT
A scalar quantity is a physical quantity that can be completely specified by a
single number together with an appropriate unit of measurement. In short, it is a
quantity with magnitude only. For instance, it makes perfectly good sense to say that
the length of an object is 1.42 m or that the mass of an object is 12.2 kg. Quantities
that can be specified in this simple and straightforward way are called scalar
quantities. Scalar quantities are often referred to simply as scalars.
What are the different scalar quantities?
Table 2.1: Scalar quantities
Scalar
Unit
Distance
Meter
Speed
Meter/second
Work
Joules
Energy
Joules
Density
Kilogram/cubic meter
Volume
Cubic meter
Scalar
Specific Heat
Entropy
Temperature
Charge
Frequency
Unit
Joules/Kg
Joules/Kelvin
, ,
Coulomb
Hertz
What is a Vector quantity?
A vector quantity or simply vector is a physical quantity that can be
completely specified by a magnitude and direction. The general “magnitude” term is
referred to here is always a non-negative scalar quantity. The magnitude of any vector
can be thought of as the „size‟ or „length‟ of that vector. This is represented by an arrow
whose length represents the magnitude (how far, how strong etc, depending on the
type of vector) and the arrow head represents the direction which is often specified by
an angle. See illustration.
N
Figure 2.1: Vector representation
The magnitude of the vector in the given illustration is 30 N and its direction is
to the East or to the right.
8
What are the different vectors quantities?
Table 2.2: Vector quantities
Vector
Unit
Displacement
Meter
Velocity
Meter/second
Force
Newton
Acceleration
Meter/square second
Weight
Newton
Momentum
Kilogram-meter/second
Shearing stress
Pascal
Electric field intensity
Volts/meter
Magnetic field intensity
Tesla
Adding and Subtracting Vectors
Two vector quantities of the same type (e.g. two displacements) may be added
together to produce another vector quantity of the same type. This operation is called
vector addition. It is straightforward, though quite different from the everyday
operation of addition with which you are familiar.
Before examining the general rules for vector addition let‟s look at an example
to see how the process works in practice. Figure 2.2 shows the points O, P, Q, R and
S we shall use these points to provide the examples we need.
Figure 2.2: The points O, P, Q, R, S and their locations in the (x, y) plane.
Imagine yourself to be located at Q (see Figure 2.2), the lowest of the five points,
and suppose that you undergo a displacement ⃗⃗⃗⃗⃗⃗ . Where would you find yourself?
Obviously, at O. Now suppose you undergo another displacement ⃗⃗⃗⃗⃗ . What would your new
location be after this second displacement? You would find yourself at point S. So the
overall result of the two successive displacements ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗ would be to move you
from Q to S. But, of course, such an effect could also have been produced by the
⃗⃗⃗⃗ . Thus, it makes sense to write
⃗⃗⃗⃗
⃗⃗⃗⃗
Equation (1)
single displacement
⃗⃗⃗⃗
9
The left – hand side of Equation 1 contains a new kind of quantity – the sum of
two vectors. Thus Equation 1 is an example of vector addition and the vector ⃗⃗⃗⃗⃗ on the
right – hand side is said to be the vector sum or resultant of ⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ . The magnitude
of the vector sum or resultant can be calculated using the Pythagorean Theorem
a
c
𝛉
b
where is the resultant and
are the two vectors. The direction of the resultant
can be calculated using the basic trigonometric function
Examples: Refer to Figure 2.2.
Complete the following vector sums.
1.
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
2.
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
3.
(⃗⃗⃗⃗⃗
4.
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ ) ⃗⃗⃗⃗⃗⃗
(⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗ )
(⃗⃗⃗⃗⃗ )
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
(⃗⃗⃗⃗⃗ )
⃗⃗⃗⃗⃗
So far, all the examples of vector addition that we have considered have involved
vectors that are „nose – to – tail‟ like ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ , but remember, a vector is completely
specified by its magnitude and direction; the endpoints are insignificant. Points such
as O, P, Q, R and S merely provide a convenient way of specifying vectors, the vectors
themselves could just as easily be represented by bold – face letters, such as the u and
v shown in Figure 2.3 together with their sum w. Moreover, any vector that has the
same magnitude and direction as v is equal to v, irrespective of its endpoints, and
could be used in place of v in the equation
Equation (2)
Figure 2.3: Two vectors u and v together with their sum w.
Figure 2.4 shows just such a vector, t. Even though t is shown in a different location
from v, the fact that t v means that it is correct to write
Equation (3)
10
Figure 2.4: The vectors v and t are equal.
So what is the general rule for adding two vectors?
The answer is illustrated graphically in Figure 2.5 and may be summarized in
the following triangle rule for vector addition.
The Triangle Rule
Let vectors a and b be
represented by appropriate arrows (or
directed line segments). If the arrow
representing b is drawn from the head
of the arrow representing a, then an
arrow from the tail of a to the head of b
represents the vector sum a b,
marked c in the figure.
F
Figure 2.5: The Triangle Rule
Examples: Refer to Figure 2.6.
Sketch and label some simple diagrams showing how the Triangle Rule can be used to
find the following vector sums.
a.
b.
c.
Figure 2.6: Vector Addition
Answers:
a.
b.
B
c.
D
𝐆
E
A
C
𝐄
𝐅
F
11
An alternative but equivalent method of adding vectors graphically is provided
by parallelogram rule. This has no real advantages over the triangle rule, but it is
preferred by some authors. It is illustrated in Figure 2.7 and may be stated as follows.
The Parallelogram Rule
Let vectors a and b be
represented by appropriate arrows (or
directed line segments). If the arrows
representing a and b are drawn from a
common point O so they form two
sides of a parallelogram, when the
parallelogram is completed an arrow
from O along the diagonal of the
parallelogram represents the vector
sum 𝐚 𝐛, marked c in the figure.
Figure 2.7: The Parallelogram Rule
Despite the lengthy discussion of vector addition, nothing has yet been said
about vector subtraction. The time has come to remedy the deficiency. Look at Figure
2.8 carefully.
Figure 2.8: (a) Two vectors (b) Vector Sum (c) Vector Difference
Given two vectors of the same type, such as the vectors a and b shown in
Figure 2.8a, you already know how to add them together to form their sum a
b
( )
(Figure 2.8b). Figure 2.8c is the summation of vectors a and –b or
This quantity is usually more written as
and is called the Vector Difference of a
and b.
Components of Vectors
Imagine a ball released from rest on a
perfectly smooth inclined plane, as shown in Figure
2.9a. What will happen to the ball immediately after
release? Obviously, the ball will start to move down
the plane, accelerating as it does so. Anyone
familiar with Newton‟s Laws of Motion would say
that the acceleration of the ball must be caused by
a force pointing down the plane. But what is the
origin of the force causing the acceleration? The
only „downward‟ force that acts on the ball is its
weight, W – the force that arises from the action of
gravity on the ball‟s mass – and that force acts
vertically downwards, not parallel to the plane. So
where does the accelerating force come from?
Figure 2.9. (a) A ball of weight W, released from rest on an inclined plane. (b) The
component vectors of W, parallel and normal to the plane.
12
Happily, vector addition provides a simple
answer. The weight W of the ball can be regarded as
the sum of two other forces as shown in Figure
2.9b, and we can write
. The force
that
is parallel to the plane causes the acceleration,
while the force , that is normal (at right angles) to
the plane stops the ball from leaving the plane and
accounts for the difference between W and .
This process of splitting a given vector into
constituent parts at right angles to each other is
called (Orthogonal) Resolution; we speak of
resolving the vector into its (orthogonal) Component
Vectors along the chosen directions.
Figure 2.10. A given vector a and a line AB. The
vector is to be resolved into component vectors parallel (ap) and normal (an) to the line AB.
One problem you might be asked to solve is illustrated in Figure 2.10. A vector
a is given, and a line AB, inclined at an angle
to a, is specified. The problem is to
resolve vector a into two component vectors, one parallel to AB, the other normal to
AB. To solve the problem, just use the parallelogram rule („rectangle rule‟). Construct a
rectangle like the one shown in Figure 2.10, with a as its diagonal and one side
parallel to AB. Call the component vectors parallel and normal to AB, respectively, ap
and an. Applying basic trigonometry to the rectangle, you should then be able to see
that the magnitudes of the two orthogonal component vectors are:
Equation (4)
Equation (5)
Vectors can also be resolved into components such as the X-component (or
horizontal component) and the Y-component (or vertical component) relative to the
Cartesian plane and the equations will be the following:
or
Equation (6)
or
Equation (7)
Examples:
A. Resolving vectors into component vectors relative to the given lines:
1.
𝐀
𝐁
𝐀
𝐁
Solution:
𝑎𝑛
𝑎𝑝
13
2.
𝐀
𝐁
𝐀
𝐁
Solution:
𝑛
𝐛
𝟓𝟎 𝐍
𝑝
Alternative Solution:
𝐜
3.
𝟏𝟖 𝐦/𝐬
𝐁
𝟔𝟎
𝐀
Solution:
𝑛
𝐜
𝟏𝟖 𝐦/𝐬
𝟔𝟎
𝐁
𝑝
𝐀
14
/
/
B.
4.
Resolving vectors into components using the Cartesian plane shown.
⃗⃗⃗⃗⃗
5.
⃗⃗⃗⃗
Solution:
4. Vector ⃗⃗⃗⃗⃗ is a displacement (in meters) from
point O to P. Its vector components are ⃗⃗⃗⃗⃗⃗⃗ and
⃗⃗⃗⃗⃗⃗⃗ . ⃗⃗⃗⃗⃗⃗⃗ has a measurement of 2 m directed to
the East or to the right while ⃗⃗⃗⃗⃗⃗⃗ has a
measurement of 2 m also directed North or
upward. Thus the vector components are:
⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗
5. Vector ⃗⃗⃗⃗ is a displacement (in meters) from
point S to R. Its vector components are ⃗⃗⃗⃗⃗⃗⃗ and
⃗⃗⃗⃗⃗⃗⃗ . ⃗⃗⃗⃗⃗⃗⃗ has a measurement of 2 m directed to
the West or to the left while ⃗⃗⃗⃗⃗⃗⃗ has a
measurement of 4 m directed North or upward.
Thus the vector components are:
⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗
15
C.
6.
Calculating the magnitude and direction of the resultant vector:
An airplane flying east at
/ has a
/ tailwind. What is the resultant
velocity of the plane?
𝟒𝟓 𝐦/𝐬
𝟑𝟏𝟎 𝐦/𝐬
Solution:
Magnitude Calculation:
/
Angle Calculation:
Resultant Velocity:
⁄
7.
You left your house to visit a friend. You got in your car, drove 40 miles east,
then got on a highway and went 50 miles north. What is your resultant vector?
Solution:
𝟓𝟎 𝐦𝐢𝐥𝐞𝐬
𝛉
𝟒𝟎 𝐦𝐢𝐥𝐞𝐬
Magnitude Calculation:
Using Pythagorean Theorem:
√
√(
)
(
)
√
miles
Angle Calculation:
(
)
Resultant Velocity:
16
8.
An airplane flies at
North of East with a velocity of
/ . It then flies
toward North with a velocity of
/ . What is the resultant velocity of the
airplane?
/
/
𝟑𝟎
Solution:
This problem involves both the concepts of resultant vector and components of
vectors.
𝐕𝒚
𝐕𝟐𝒚
𝐕𝟐𝒙
𝟏𝟓𝟎 𝐤𝐦/𝐡𝐫
𝟎
𝐕
𝛉
𝐕𝟏𝒚
𝟑𝟎
𝟐𝟎𝟎 𝐤𝐦/𝐡𝐫
𝐕𝟏𝒙
𝐕𝒙
Vector
X Component
Y Component
In this table, all horizontal components will be added same is true with the
vertical components. Thus the horizontal component of the resultant velocity is
and its vertical component is
. Then apply Pythagorean
Theorem to get the value of the magnitude of the resultant velocity.
Magnitude Computation:
or
√
√(
)
(
)
17
Angle Computation:
(
)
Resultant Velocity:
Points to Consider:
When adding or subtracting two vectors or more, directions of these vectors
should be considered. For example, if two vectors are given with the same direction,
we add the magnitude of these vectors and carry out the direction. But if one of these
vectors has an opposite direction, we subtract the vector with lesser magnitude from
the vector with greater magnitude and carry out the direction of the greater vector to
be the direction of the resultant vector. Look at the examples on the next page.
⁄
⁄
⁄
⁄
/
/
/
/
/
/
⁄
⁄
⁄
/
/
/
⁄
/
/
/
18
IT’S YOUR TURN
Exercise 2.1.
For letters A & B, refer to the given figure. Units are in meters.
A.
Complete the following vector sums. (
⃗⃗⃗⃗⃗ ⃗⃗⃗⃗
1.
____
2.
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
____
3.
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
____
4.
(⃗⃗⃗⃗
⃗⃗⃗⃗⃗
5.
B.
C.
⃗⃗⃗⃗⃗ )
(⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
____
⃗⃗⃗⃗⃗ )
____
)
Resolve the following vectors into components. Sketch the diagram of each.
(
)
⃗⃗⃗⃗⃗
6.
⃗⃗⃗⃗⃗
7.
Calculate the magnitude and direction of the resultant vector. Draw the diagram.
8.
A man walks
from A to B at
North of East, and then walks
from B to C due East. How far and at what angle is the man‟s final position
from his initial position? (
)
B
C
A
19
LESSON 3: LINEAR MOTION
Objectives:
At the end of the lesson, you should be able to:
1.
determine the displacement of a moving body in linear motion;
2.
calculate the velocity of a moving body in linear motion; and
3.
compute the acceleration of a moving body in linear motion.
LET’S ENGAGE
How can you tell if an object is moving?
Everything moves. Even things that appear to be at rest move. They move with
respect to the sun and stars. When we describe the motion of one object with respect
to another, we say that the object is moving relative to the other object. A book that is
at rest, relative to the table it lies on, is moving at about 30 kilometers per second
relative to the sun. The book even moves faster relative to the center of our galaxy.
When we discuss the motion of something, we describe its motion relative to
something else. An object is moving if its position relative to a fixed point is
changing. When we say that a space shuttle moves at 8 kilometers per second, we
mean its movement relative to Earth below. When we say a racing car reaches a speed
of 300 kilometers per hour, of course we mean relative to the track. Unless stated
otherwise, when we discuss the speeds of things in our environment, we mean speed
with respect to the surface of Earth even though Earth moves around the sun.
Think!
A hungry mosquito sees you resting in a hammock in a 3 – meter per second
breeze. How fast and in what direction should the mosquito fly in order to
hover above you for lunch?
Answer : (𝒔𝒆𝒆 𝟑 𝟏)
LET’S TALK ABOUT IT
What is Speed?
Before the time of Galileo, people described moving things as simply “slow” or
“fast”. Such descriptions were vague. Galileo is credited as being the first to measure
speed by considering the distance covered and the time it takes. Speed is how fast an
object is moving. You can calculate speed of an object by dividing the distance covered
by time.
For example, is a cheetah, such as the one shown in Figure 3.1 covers 50
meters in a time of 2 seconds, its speed is
/ .
20
(Hewitt, 2005)
Figure 3.1. Cheetah is the fastest land animal over distances less than
meters and
can achieve peak speeds of
/ .
Any combinations of units for distance and time that are useful and convenient
are legitimate for describing speed. Miles per hour (mi/h), kilometres per hour (km/h),
centimeters per day (cm/day), or light – years per century are all legitimate units for
speed. The slash symbol (/) is read as “per”. Throughout this subject course, we‟ll
primarily use the unit meters per second ( / ) for speed.
Think!
If a cheetah can maintain a constant speed of
every second. At this rate, how far will it travel in
Answer : (𝒔𝒆𝒆 𝟑 𝟐)
How can you calculate speed?
A car does not always move at the
same speed. A car may travel down a street
at
/ , slow to
/ at a red light,
and speed up to only
/ because of
traffic. You can tell the speed of the car at
any instant by looking at the car‟s
speedometer, such as the one in Figure
3.2. The speed at any instant is called the
instantaneous speed. A car traveling at
/ may go at that speed for only one
minute. If the car continued at that speed
for a full hour, it would cover
. If it
continued at that speed for only half an
hour, it would cover only half that
distance, or
. In one minute, the car
would cover less than 1 km.
/ , it will cover
meters
seconds? In 1 minute?
(Hewitt, 2005)
Figure 3.2. The speedometer for a North
American
car
gives
readings
of
instantaneous speed in both 𝑚𝑖/ and
𝑘𝑚/ . Odometers for the U.S. market give
readings in miles; those for the Canadian
market give readings in kilometres.
21
In planning a trip by car, the driver often wants to know how long it will take to
cover a certain distance. The car will certainly not travel at the same speed all during
the trip. The driver only cares about the average speed for the trip as a whole. The
average speed is the total distance covered divided by the time.
Average speed can be calculated rather easily. For example, if we drive a
distance of 60 kilometers during a time of 1 hour, we say our average speed is 60
kilometers per hour (60 km/h). Or, if we travel 240 kilometers in 4 hours,
/
Note that when a distance in kilometers is divided by a time in hours, the
answer is kilometers per hour (km/h).
Since average speed is the distance covered divided by the time of travel, it does
not indicate variations in the speed that may take place during the trip. In practice, we
experience a variety of speeds on most trips, so the average speed is often quite
different from the instantaneous speed. Whether we talk about average speed or
instantaneous speed, we are talking about the rates at which distance is travelled.
Think!
The speedometer in every car also has an odometer that records the distance
travelled. If the odometer reads zero at the beginning of a trip and 35 km a half
hour later, what is the average speed?
Answer : (𝒔𝒆𝒆 𝟑 𝟑)
How is velocity different from speed?
In every language, we use the words speed and velocity interchangeably. In
physics, we make a distinction between the two. Very simply, the difference is that
velocity is speed in a given direction. When we say a car travels at
/ , we are
specifying its speed. But if we say a car moves at
/ to the north, we are
specifying its velocity. Speed is a description of how fast an object moves; velocity
is how fast and in what direction an object moves.
A quantity such as velocity that specifies direction as well as magnitude is
called vector quantity. Recall in Lesson 1 that quantities that require only magnitude
for a description are scalar quantities. Speed is a scalar quantity. Velocity, like force,
is a vector quantity.
Constant velocity means steady speed. Something with constant speed doesn‟t
speed up or slow down. Constant velocity, on the other hand, means both constant
speed and constant direction. Constant direction is a straight line – the object‟s path
doesn‟t curve. So, constant velocity means motion in a straight line at constant speed.
The car in Figure 3.3 may have a constant speed but its velocity is changing.
22
(Hewitt, 2005)
Figure 3.3. The car on the circular track may have a constant speed but not a constant
velocity, because its direction of motion is changing every instant.
If either the speed or the direction (or both) is changing, then the velocity is
changing. Constant speed and constant velocity are not the same. A body may move at
constant speed along a curved path, for example, but it does not move with constant
velocity, because its direction is changing every instant.
In a car there are three controls that are used to change the velocity. One is the
gas pedal, which is used to maintain or increase the speed. The second is the break,
which is used to decrease the speed. The third is the steering wheel, which is used to
change the direction.
Think!
The speedometer of a car moving northward reads
/ . It passes another
car that travels southward at
/ . Do both cars have the same speed? Do
they have the same velocity?
Answer : (𝒔𝒆𝒆 𝟑 𝟒)
What is acceleration?
We can change the state of motion of an object by changing its speed, its
direction, or both. Any of these changes is a change in velocity. Sometimes we are
interested in how fast the velocity is changing. A driver on a two – lane road who
wants to pass another car would like to be able to speed up and pass in the shortest
possible time. Acceleration is the rate at which the velocity is changing. You can
calculate the acceleration of an object by dividing the change in velocity by time.
We are familiar with acceleration in an automobile, such as the one shown in
Figure 3.4. The driver depresses the gas pedal, which is appropriately called the
accelerator. The passengers then experience acceleration, or “pick up” as it sometimes
called, as they are pressed into their seats. The key idea that defines acceleration is
change. Whenever we change our state of motion, we are accelerating. A car that can
accelerate well has the ability to change its velocity rapidly. A car that can go from
zero to 60 km/h in 5 seconds has a greater acceleration than another car that can go
80 km/h in 10 seconds. So having good acceleration means being able to change
23
velocity quickly and does not necessarily refer to how fast something is moving. In
physics, the term acceleration applies to decreases as well as increases in speed. The
brakes of a car can produce large retarding acceleration, that is, they can produce a
large decrease per second in the speed. This is often called deceleration. We experience
deceleration when the driver of a bus or car slams on the brakes and we tend to hurtle
forward.
(Hewitt, 2005)
Figure 3.4. A car is accelerating whenever there is a change in its state of motion.
Acceleration also applies to changes in direction. If you ride around a curve at a
constant speed of
/ , you feel the effects of acceleration as your body tends to
move toward the outside of the curve. You may round the curve at constant speed, but
your velocity is not constant, because your direction is changing every instant. Your
state of motion is changing: you are accelerating. It is important to distinguish
between speed and velocity. Acceleration is defined as the rate of change in velocity,
rather than speed. Acceleration, like velocity, is a vector quantity because it is
directional. The acceleration vector points in the direction the velocity is changing as
shown in Figure 3.5. If we change speed, direction, or both, we change velocity and we
accelerate.
(Hewitt, 2005)
Figure 3.5. When you accelerate in the direction of your velocity, you speed up; against
your velocity, you slow down; at an angle to your velocity, your direction changes.
When straight line motion is considered, it is common to use speed and velocity
interchangeably. When the direction is not changing, acceleration may be expressed as
the rate at which speed changes.
24
Speed and velocity are measured in units of distance per time. Since
acceleration is the change in velocity or speed per time interval, its units are those of
speed per time. If we speed up, without changing direction, from 0 to 10 km/h in 1
second, our change in speed is 10 km/h in a time interval of 1 second. Our
acceleration along a straight line is
/
/
The acceleration is 10 km/h s, which is read as “10 kilometers per hoursecond”. Note that a unit for time appears twice: once for the unit of speed and again
for the interval of time in which the speed is changing.
Think!
In 5 seconds a car moving in a straight line increases its speed from
/ to
/ , while a truck goes from rest to
/ in a straight line. Which
undergoes greater acceleration? What is the acceleration of each vehicle?
Answer : (𝒔𝒆𝒆 𝟑 𝟓)
Answers for Think:
3.1: The mosquito should fly toward you into the breeze. When above you it should fly
at 3 meters per second in order to hover at rest above you. Unless its grip on your skin
is strong enough after landing, it must continue flying at 3 meters per second to keep
from being blown off. That‟s why a breeze is an effective deterrent to mosquito bites.
3.2: In 10 seconds the cheetah will cover 250 m, and in 1 minute (or 60 s) it will cover
1500 m.
3.3:
3.4: Both cars have the same speed, but they have opposite velocities because they
are moving in opposite directions.
3.5: The car and truck both increase their speed by 15 km/h during the same time
interval, so their acceleration is the same.
IT’S YOUR TURN
Exercise 3.1.
1.
How far will you travel if you maintain an average speed of
/ for
seconds?
2.
Find your average speed if you run
meters in
seconds.
⁄
3.
What is the acceleration of a car (in
) moving along a straight line path that
increases its velocity from
/ to
/ in
seconds?
⁄ for 5 seconds. What is
4.
A car going at
/ undergoes an acceleration of
its final speed (
/ )?
25
LESSON 4: FREE FALL
Objectives:
At the end of the lesson, you should be able to:
1.
calculate the distance travelled by an object thrown upward or downward;
2.
calculate the initial velocity of an object thrown upward or downward;
3.
calculate the final velocity of an object thrown upward or downward; and
4.
calculate the time of an object thrown upward or downward at a certain
distance or velocity.
LET’S ENGAGE
If an apple falls from a tree, does it accelerate while falling?
We know that the apple starts from a rest position and gains speed as it falls.
We know this because it would be safe to catch if it fell a meter or two, but not if it fell
from a high-flying balloon. Thus, the apple must gain more speed during the time it
drops from a great height than during the shorter time it takes to drop a meter. This
gain in speed indicates that the apple does accelerate as it falls.
LET’S TALK ABOUT IT
Falling Objects
Gravity causes the apple to accelerate downward once it begins falling. In real
life, air resistance affects the acceleration of a falling object. Let‟s imagine there is no
air resistance and that gravity is the only thing affecting a falling object. An object
moving under the influence of the gravitational force only is said to be in Free Fall.
Free falling objects are affected only by gravity. Table 4.1 shows the instantaneous
speed at the end of each second of fall of a freely falling object dropped from rest. The
elapsed time is the time that has elapsed, or passed, since the beginning of any
motion, in this case the fall.
Note in Table 4.1 the way the speed changes. During each second of fall, the
instantaneous speed of the object increases by an additional 10 meters per second.
This gain in speed per second is the acceleration.
Table 4.1. Free fall speeds of objects
Elapsed Time (seconds)
Instantaneous Speed (m/s)
0
0
1
10
2
20
3
30
4
40
5
50
t
10t
The acceleration of an object in free fall is about 10 meters per second squared
⁄
(
). For free fall, it is customary to use the letter g to represent the acceleration
because the acceleration is due to gravity. Although g varies slightly in different parts
⁄ . More accurately, g is
⁄ , but it
of the world, its average value is nearly
2
is easier to see the ideas involved when it is rounded off to 10 m/s . Where accuracy is
⁄ should be used for the acceleration during free fall.
important, the value of
26
Note in Table 4.1 that the instantaneous speed of an object falling from rest is equal to
the acceleration multiplied by the amount of time it falls, the elapsed time.
The letter V symbolizes both speed and velocity. Take a moment to check this
⁄ is
equation with Table 4.1.You will see that whenever the acceleration
multiplied by the elapsed time in seconds, the result is the instantaneous speed in
meters per second.
The average speed of any object moving in a straight line with constant
acceleration is calculated the way we find the average of any two numbers: add them
and divide by 2. For example, the average speed of a free falling object in its first
second of fall is the sum of its initial and final speed, divided by 2. So, adding the
⁄ , and then dividing by 2, we get
⁄ .
initial speed of zero and the final speed of
Average speed and instantaneous speed are usually very different.
Discover!
Can you catch a falling bill?
1.
Have a friend hold a peso bill so the
midpoint hangs between your fingers.
2.
Have your friend release the bill
without warning. Try to catch it.
3.
Think. How much reaction time do you
have when your friend drops the bill?
So far, we have been looking at objects moving straight
downward due to gravity. Now consider an object thrown
straight up. It continues to move upward for a while, then it
comes back down. At the highest point, when the object is
changing its direction of motion from upward to downward, its
instantaneous speed is always zero. It then starts downward
just as if it had been dropped from rest at that height. Note that
when an object is thrown upward, it is always propelled upward
with an initial velocity of some value and its motion is subject
only to gravitational effects after being released.
During the upward part of this motion, the object slows
from its initial upward velocity to zero velocity. We know the
object is accelerating because its velocity is changing. How
much does its speed decrease each second? It should come as
no surprise that the speed decreases at the same rate it
increases when moving downward – at 10 meters per second
each second. So as Figure 4.1 shows, the instantaneous speed
at points of equal elevation in the path is the same whether the
object is moving upward or downward. The velocities are
different of course, because they are in opposite directions.
During each second, the speed or the velocity changes by 10
m/s. The acceleration is 10 m/s2 downward the entire time,
whether the object is moving upward or downward.
Figure 4.1. The change in speed each second is the same whether
the ball is going upward or downward.
(Hewitt, 2005)
27
For a falling object, how does the distance per second change?
How fast something moves is entirely different from how far it moves – speed
and distance are not the same thing. To understand the difference, return to Table
4.1. At the end of the first second, the falling object has an instantaneous speed of
/ . Does this mean it falls a distance of 10 meters during this first second? No.
Here‟s where the difference between instantaneous speed and average speed come in.
The initial speed of the fall is zero and takes a full second to get to
/ . So the
average speed is halfway between zero and
/ – that‟s
/ , as discussed earlier.
So during the first second, the object has an average speed of
/ and falls a
distance of 5 m.
Table 4.2 shows the total distance moved by a free falling object dropped from
rest. At the end of one second, it has fallen 5 meters. At the end of 2 seconds, it has
dropped a total distance of 20 meters. At the end of 3 seconds, it has dropped 45
meters altogether. For each second of free fall, an object falls a greater distance
than it did in the previous second. These distances form a mathematical pattern at
the end of time t, the object has fallen a distance D of
.
Table 4.2. Free fall distances of an object
Elapsed Time (seconds)
Distance Fallen (meters)
0
0
1
5
2
20
3
45
4
80
5
125
t
We used freely falling objects to describe the relationship between distance
travelled, acceleration, and velocity acquired. In our examples, we used the
⁄ . But accelerating objects need not be freely falling
acceleration of gravity,
objects. A car accelerates when we step on the gas or brake pedal. Whenever an
object‟s initial speed is zero and the acceleration g is constant, that is, steady and
“non-jerky”, the equations for the velocity and distance travelled are
and
Think!
An apple drops from a tree and hits the ground in one second. What is its
speed upon striking the ground? What is its average speed during that one
second? How high above ground was the apple when it first dropped?
What is the relationship between velocity and acceleration?
Some of the confusion that occurs in analyzing the motion of falling objects
comes about from mixing up “how fast” with “how far”. When we wish to specify how
fast something freely falls from rest after a certain elapsed time, we are talking about
speed or velocity. The appropriate equation is
. When we wish to specify how far
that object has fallen, we are talking about distance. The appropriate equation is
. Velocity or speed (how fast) and distance (how far) are entirely different from
each other.
28
One of the most confusing concepts encountered in this subject course is
acceleration, or “how quickly does speed or velocity change.” What makes acceleration
so complex is that it is a rate of a rate. It is often confused with velocity, which is
itself a rate (the rate at which distance is covered). Acceleration is not velocity, nor is it
even a change in velocity. Acceleration is the rate at which velocity itself changes.
How are the directions of displacement, velocity, and acceleration?
As velocity, displacement, and acceleration are vectors, direction needs to be
recognized (positive or negative). To minimize confusion in the sign conventions of
these vector quantities, we will have the initial velocity of any moving object to be
always positive in value whether the direction is upward or downward. Directions of
final velocity, displacement, and acceleration will be based on the direction of the
initial velocity. So if their directions are the same with the direction of the initial
velocity, positive, but if opposite the direction of the initial velocity, negative. For
example, if the object is moving upward, the directions of the displacement, and final
velocity, and acceleration are directed upward and they are of the same direction as
the initial velocity. The only opposite in direction is the gravitational acceleration
which is always downwards, so its value will be negative. In calculating the
displacement, velocity and acceleration of an object, we can also use the three
kinematic equations
Where:
final velocity
initial velocity
gravitational acceleration
displacement
time
Answer for Discover:
Expected Outcome: It will be impossible for the friend to catch the falling peso bill
because the reaction time is greater than the time it takes the bill to fall 8 cm.
Think: Less than
of a second to catch the bill.
Answer for Think:
When it hits the ground, the apple‟s speed will be
and it starts 5 m above the ground.
/ . Its average speed is
/ ,
Examples:
1.
A stone is dropped from a 30 m tower.
a. How fast is it going when it hits the ground?
b. How long does it take to hit the ground?
Solution:
⁄ ,
a. Given:
,
Req‟d:
𝑖
⁄
𝑓
29
Since the directions of the final velocity, displacement, and gravitational
acceleration are the same with the direction of the initial velocity, thus the
values of the final velocity, displacement, and gravitational acceleration will be
positive. For objects moving downward, the equation to be used will be as
follows:
( )
(
)(
)
√
/
b. Given:
Req‟d: t
2.
,
⁄ ,
,
/
If a ball is thrown vertically upward with an initial velocity of
/ , how high
will it go?
Solution:
Gravitational acceleration is the only quantity opposite the direction of the
initial velocity, thus its value will be negative. At the maximum height of the ball,
final velocity is zero.
𝑓
⁄
/
𝑖
Given:
Req‟d: D
( )
3.
(
/ ,
)
(
⁄
,
)( )
A stone is thrown vertically upward and reached its maximum height in 5
seconds.
a. What is the initial velocity of the stone?
b. What is the maximum height of the stone?
Solution:
⁄
a. Given:
,
,
𝑓
⁄
𝑖
30
( )
( )
⁄
b. Given:
( )
(
( )
,
)
(
,
(
⁄ ,
⁄
)
)
IT’S YOUR TURN
Exercise 4.1.
1.
2.
3.
How far (
) will a free falling object fall from rest in 8 seconds?
An apple drops from a tree and hits the ground in 3.8 seconds. What is its final
speed (
/ ) when it hits the ground?
A stone was thrown vertically upward and reached its maximum height in 4.6
seconds.
a. What is the initial velocity of the stone?
b. How high will the stone go?
31
LESSON 5: PROJECTILE
Objectives:
At the end of the lesson, you should be able to:
1.
calculate the horizontal and vertical components with respect to velocity
and position of a projectile at various points along its path;
2.
determine the distance travelled by the projectile at various points along
its path; and
3.
compute the time of the projectile at various points along its path.
LET’S ENGAGE
Examine the figure on the next page. Which ball strikes the ground first, the one that
is projected horizontally or the one that is dropped freely?
(Hewitt, 2005)
Both balls fall the same vertical distance with the same acceleration and
therefore strike the ground at the same time. Analyze the curved path of the ball by
considering the horizontal and vertical velocity components of separately. There are
two important things to notice. The first is that the ball‟s horizontal component of
motion remains constant. The ball moves the same horizontal distance in the equal
time intervals between each flash, because no horizontal component of force is acting
on it. Gravity acts only downward, so the only acceleration of the ball is downward.
The second thing to note is that both balls fall the same vertical distance at the same
time. The vertical distance fallen has nothing to do with the horizontal component of
motion.
32
LET’S TALK ABOUT IT
What is Projectile?
A projectile is any object projected by some means and continues to move due
to its own inertia (mass). It acts only under the influence of gravity and follows a
parabolic path as shown in Figure 5.1.
(Simisterlucy, 2019)
Figure 5.1. Projectile
A projectile moves in two dimensions, therefore, it has two components: the
horizontal and the vertical just like a resultant vector as shown in Figure 5.2. Velocity,
displacement, and acceleration are vector quantities, thus in projectiles, they have
also the horizontal and vertical components.
𝐃
𝒚
𝒙
Figure 5.2. A projectile in two dimensions.
33
The horizontal component of velocity never changes but covers equal
displacements in equal time periods. This component is just like the horizontal motion
of a ball rolling freely along a level surface. When friction is negligible, a rolling ball
moves at constant velocity. With no horizontal force acting on the ball there is no
horizontal acceleration. The same is true for projectile – when no horizontal force acts
on the projectile, the horizontal velocity remains constant. But why? Gravity does not
work horizontally to increase or decrease the velocity. Gravity applies only to the
vertical motion of an object. This means that the horizontal component of the
acceleration is zero. Horizontal displacement of projectile is given by the equation
where is the horizontal displacement,
is the horizontal component of velocity and
t is time. This can also be calculated using the equation
where
is the maximum range or the maximum horizontal displacement,
is the
initial velocity,
is the angle of launch of the projectile and g is the gravitational
acceleration. In this equation, value of the gravitational acceleration is always positive.
For the vertical motion of the projectile, it applies all the concepts of Free Fall
since projectile motion is under the influence of gravity. Thus, all equations used in
Free Fall will be used also in Projectile. But take note that the motion is vertical so for
the velocity and displacement, vertical components shall be used. So just like in Free
Fall, at the top of the projectile‟s trajectory (path), vertical component of velocity is zero
(
). We will also use the three kinematic equations
where
is the vertical component of final velocity,
is the vertical component of
the initial velocity and D is the vertical displacement. We can also use the derived
formula for the vertical displacement at any point on the path which is
Calculation of velocity components will apply the equations used in vector
components and these equations are
Examples:
1.
An object is fired from the ground at
/ at an angle of
with the horizontal.
a. What are the horizontal and vertical components of the initial velocity?
b. How high will the object go?
c. How long will it take to reach the top of its trajectory?
34
𝐕𝒇𝒚
𝐕𝒊𝒚
𝐃
𝐠
30°
Solution:
a. Given:
Req‟d:
𝟎
𝐕𝒊𝒙
/ ,
,
/
/
b. Given:
/ ,
,
⁄
(
)
Req‟d: D
( )
(
c. Given:
Req‟d: t
2.
)
(
)( )
/ ,
,
⁄
An object is launched at a velocity of
/ in a direction making an angle of
with the horizontal.
a. What is the maximum height reached by the object?
b. What is the maximum range of the object?
c. What is the total time of flight of the object?
35
𝐕𝒇𝒚
𝟎
𝐕𝒊𝒚
𝐃
25°
𝐕𝒊𝒙
R
Solution:
a. Given:
Req‟d: D
/ ,
,
⁄ ,
;
( )
(
)
b. Given:
Req‟d:
(
(
)
/ ,
)
c. Given:
Req‟d: t
(
,
⁄
)
/ ,
;
3.
A ball is kicked at an angle of
with the ground.
a. What should be the initial velocity of the ball so that it hits a target that is 30
meters away at a height of 1.8 meters?
b. What is the time for the ball to hit the target?
𝐕𝒊
𝒚
𝒙
36
Solution:
a. Given:
Req‟d:
,
,
(
(
,
⁄
)
)
√
/
b. Given:
Req‟d: t
,
,
/ ,
IT’S YOUR TURN
Exercise 5.1:
1.
A projectile is launched from the ground with an initial velocity of
angle of
with the horizontal.
/ at an
Top of Trajectory
𝐃
20°
R
a.
b.
c.
d.
What is the horizontal component of the initial velocity? (
How long will it take to reach the top of its trajectory? (
)
How high will it go? (
What will be its maximum range? (
)
)
)
37
2.
A stone is kicked from the ground with an initial velocity of
with the horizontal.
/ at an angle
D
28°
R
a.
b.
c.
d.
What is the maximum height of the stone? (
)
How long will it take for the stone to reach its maximum height? (
What is its maximum range? (
)
How long will it take to reach the maximum range? (
)
)
38
REFERENCES
Hewitt, P. G. (2005). Conceptual physics. Pearson Education South Asia Pte Ltd.
Serway, R. A., & Faughn, J. S. (2nd ed.). (1989). College physics. Sauders College
Publishing.
Simisterlucy. (2019) Projectile motion in basketball. https://www.tes.com/teachingresources/shop/simisterlucy
The
Open
University.
(1998).
Vectors
and
scalars
pdf.
http://www.cse.salford.ac.uk/physics/gsmcdonald/pp/PPLATOResources/hflap/M2_4t.pdf
39