Topic 2 Forces and Motion

� INTRODUCTION

The motion of objects such as cars, footballs, joggers or even the motion of planets and their moons, is an obvious part of our everyday life. Our understanding of motion was established in the 16th and 17th centuries when two individuals made important discoveries: Galileo Galilei (1564-1642) and Isaac Newton (1642-1727).

By the end of this topic, you should be able to:

1. State scalar and vector quantities;

2. Find the resultant of two or more vectors;

3. Define displacement, speed, velocity and acceleration;

4. Describe the graphs of motion;

5. Solve problems related to displacement, speed, velocity and acceleration;

6. Define Newton Laws of Motion.

7. Relate mass to inertia;

8. Use the Principle of Conservation of Momentum to solve problems;

9. Solve problems related to collisions; and

10. Solve problems related to projectile motion.

LEARNING OUTCOMES

TTooppiicc

22 � Forces and

Motion

� TOPIC 2 FORCES AND MOTION

44

Mechanics is the field of physics that studies motion of objects. Mechanics are divided into two parts called kinematics, which is a description of how objects move, and dynamics, which describes movements of objects in relation to the forces acting on them. Several concepts in kinematics and dynamics and their application in daily life situations will be discussed in this topic.

VECTORS

We have learnt that a physical quantity has a value and a unit attached to it. This is called the mmagnitude of a physical quantity. However, for some quantities, it makes more sense if the directions are given as well. For example, if you want to describe the movement of a car, stating its speed alone in not enough. It will give a clearer picture when you describe the direction as well. Thus physical quantities can be divided into sscalar quantity and vvector quantity based on the information given.

2.1.1 Scalar Quantity

Scalar quantity means that the physical quantity only has mmagnitude but no direction. Examples of scalar quantities are distance, speed, mass and volume. Example 2.1: Distance = 200 m. Distance is a scalar quantity, 200 is a magnitude and m is a unit.

2.1.2 Vector Quantity

A physical quantity that has both mmagnitude and direction is called a vector quantity. Examples of vector quantities are displacement, velocity, acceleration and momentum. Example 2.2: Velocity = 70 kmh-1 to the east Velocity is a vector quantity, 70 is a magnitude mkmh-1 and Âto the eastÊ is the direction.

2.1

TOPIC 2 FORCES AND MOTION �

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2.1.3 Resultant and Resolution of Vector

An example of a vector quantity which has both magnitude and direction is force. A vector quantity is represented by a long arrow whereby the length of the arrow shows the magnitude of the force and the head shows the direction of the force (see Figure 2.1).

Figure 2.1: Addition of vector

The force obtained from the addition of two or more forces is called the rresultant force. Additions of vector quantities such as force must take into account both the direction and magnitude.

For vectors which are parallel (whether in the same direction or in opposite directions), the resultant vector can be determined by the sum of every vector present. Using Figure 2.1, the resultant vector, RR = AA + BB. If the second vector is in the opposite direction, RR = AA � BB. For non-parallel vector quantities, the resultant vector can be determined using the triangle of forces or the parallelogram of forces.

A single vector can be resolved into two components. This is known as the resolution of vectors.

Figure 2.2 shows a single vector FF resolved into its two perpendicular components, FFx and FFy.

Figure 2.2: Resolution of vectors


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The horizontal component is Fx = F Cos �

and the vertical component is Fy = F Sin � Example 2.3: A man pulls a sack of fruits with a force of 140N at an angle of 35À with the floor. Determine the horizontal and vertical component of the force. Solution: Calculate the horizontal component of the force that causes the sack to be pulled forward. Thus, cos 35À is considered. Horizontal component, Fx = 140 cos 35À Calculate the vertical component of the force that causes the sack to be pulled forward. Thus, sin 35À is considered. Vertical component, Fy = 140 sin 35À

KINEMATICS

In this subtopic, we will look at displacement, speed, velocity and acceleration. This is followed by linear motion, graphs of motion and equation of motion.

2.2

To test your understanding, consider the following quantities listed below. Categorise each quantity as being either a vector or scalar quantity:

(a) 5 m

(b) 30 ms-1 to the East

(c) 20 degrees Celcius

(d) 256 pound

(e) 5 miles North

ACTIVITY 2.1


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2.2.1 Displacement, Speed, Velocity and Acceleration

Imagine you are visiting a friend who lives several kilometres away from your home. Whichever route you choose, the starting point and finishing point remain unchanged (Figure 2.3).

Figure 2.3: Displacement versus distance travelled along a path

Your ddisplacement is the distance directly from the starting point, A directly to the finishing point, B. So whatever route you take, your displacement from A to B remains unchanged. Thus displacement can be defined as the cchange in position of the object. Displacement is a quantity that has both magnitude and direction, so it is a vvector quantity. Distance refers to the length of the path taken. It scalar quantity. (a) Speed Speed is a sscalar quantity. It describes the magnitude of how fast or how

slow an object is moving. Speed is defined as the rrate of change of distance travelled with time.

Distance travelled

Speed =Time

A car moving along a winding road or a circular track at 80km h-1 is said to

have a speed of 80 kmh-1. Speed is a quantity that has no direction but only magnitude.


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The standard unit for speed is metre per second or m s-1. Conversion of units to kilometre per hour (km h-1) and centimetres per second (cm s-1) are also commonly used.

(b) Velocity Velocity is a vvector quantity. Thus it involves both the magnitude and

direction of the moving object. Velocity is derived from displacement of an object rather than its distance. It is defined as the rrate of change of displacement or

Displacement

Speed =Time

The car in Figure 2.4 has a constant speed of 80km hr-1 as it moves along

the circular track. At every point of the track, such as P and Q, the speed is the same but velocity v1 and v2 are different. This is because the directions of the car at P and Q are different as the arrow points at different directions.

Figure 2.4: Velocity and speed

Velocity can also be measured in the units metre per second (m s-1),

kilometre per hour (km h-1) or centimetres per second (cm s-1).


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(c) AAcceleration When you reach a highway when driving, you will hit the gas so that the

car will move at a higher velocity. The rate of change of velocity is called acceleration.

Velocity change

Acceleration =Time taken

Acceleration is a vvector. The direction of acceleration is the direction of the velocity change. The unit for acceleration is metre per second per second or ms-2.

If a car accelerates from 15 ms-1 to 35 ms-1 in 5 s, then the acceleration

1

2

(35 15) msa =

5s

= 4 ms

�

�

�

On the other hand, if the car breaks, the velocity may decrease from 30 ms-1

to 20 ms-1 in 5 s. It has a retardation or deceleration of

1

2

(30 20) msa =

5 s

= 2m s

�

�

�

�

The value -2 ms-2 is a negative acceleration or deceleration.

2.2.2 Linear Motion

Linear motion means the movement of an oobject in a straight line. Examples of linear motions are, a car moving in a straight line, a train moving on a straight track, a falling coconut and a moving bullet. In linear motion, vector quantities are treated much like scalar quantity since the direction remain unchanged. Examples of non-linear motions would be a snake crawling and a roller coaster ride.

An object moves from rest with a uniform acceleration of 2 ms-2. Find the velocity of the object after 30 s.

ACTIVITY 2.2


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It can be uniform, that is, with constant velocity or non-uniform, that is, with a variable velocity. Non-uniform motion are further divided into other types of motions such as constant acceleration motion.

2.2.3 Graphs of Motion

The motion of an object could be represented in a graph to aid understanding. These would be the displacement-time graph and the velocity-time graph. From the graphs for linear motion, we can determine:

� The displacement of an object at a specific time;

� The velocity and acceleration of an object; and

� Changes in velocity and displacement at a certain time. (a) DDisplacement-Time Graph A displacement against time graph allows us to interpret movement from

the shape of the graph. Figure 2.5 shows displacement-time graphs that describe several movements of an object. Consider the gradient of the graphs as the velocity of the object.

Figure 2.5: Examples of displacement-time graphs


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The following describes further the meaning of these graphs:

(i) Figure 2.5(a): As time increases, distance is always the same. Object is not moving. Gradient of the graph is zero.

(ii) Figure 2.5(b): The distance increases as time increases. Object is moving with a constant velocity because the gradient is constant.

(iii) Figure 2.5(c): The distance increases as time increases, gradient of the graph also increasing. Object is moving with increasing velocity.

(iv) Figure 2.5(d): The distance increases as time increases, gradient of the graph decreases. Object is moving with a decreasing velocity.

(b) VVelocity-Time Graph

A velocity-time graph shows how the velocity of an object changes with time. TThe gradient of a velocity-time graph represents the acceleration of an object. We can also calculate the distance travelled by the object by calculating the area under the velocity-time graph.

Figure 2.6: Examples of velocity-time graphs


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The following describes these graphs further: (i) Figure 2.6(a): As time increases, velocity is always the same. Object

is moving with a constant velocity. Gradient of the graph is zero.

(ii) Figure 2.6(b): The velocity increases as time increases. Object is

moving with a constantly increasing velocity, or constant acceleration. The gradient of the graph, that is, the acceleration is constant.

(iii) Figure 2.6(c): The velocity increases as time increases, gradient of

the graph also increases. Object is moving with increasing acceleration.

(iv) Figure 2.6(d): The distance increases as time increases, gradient of

the graph decreases. Object is moving with a decreasing acceleration.

Example 2.4: A car is moving along a straight road. Its movement is shown in Figure 2.7.

Figure 2.7: A velocity-time graph of a car

(a) Describe the movement of the car along the straight road.

(b) Find the acceleration of the car at OA, AB and BC.

(c) Find the total distance travelled.


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Solution: (a) Along OA, the car is moving with an increasing velocity. The acceleration is

constant as the gradient along OA is constant. Along AB, the car is moving with a constant velocity. The acceleration is

zero as the gradient at AB is zero. Along BC, the car is moving with a decreasing velocity. The acceleration is

constant but has a negative value. The car is decelerating. (b) The acceleration of the car in a velocity-time graph is the gradient.

The mathematical formula for gradient is �

�2 2

2 2

=y y

mx x

OA

1

2�

��

�OA

(15 0)m sm = = 0.75m s

(20 0)s

The car is accelerating at 0.75m s-2. AB

1

3

��

�AB

(15 15)m sm = = 0

(50 0)s

The car is not accelerating. It is moving with a constant velocity. BC

1

2

5

��

��BC

(0 15)m sm = = 1.5

(60 0)sm s

The car is decelerating at 1.5m s-2. (c) To find the distance travelled in a velocity-time graph, find the area under

the graph. This total distance travelled by the car is: Total area under the graph = �(15 � 20) + (50 � 30)15 + �(60 � 50)15 = 675m


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2.2.4 Equation of Motion

Imagine a Formula 1 race car driving with an initial velocity, uu, accelerates with a constant acceleration, a, and achieves a final velocity, v. The displacement of the race car is ss. The four variables (u, vv, aa, ss) are different but related. The relationship of these four variables with one another can be shown in the equations of linear motion. From the definition of acceleration

�=

v ua

t

Where a = acceleration, u = initial velocity, v = final velocity, t = time. Hence v = u + at (1)

Displacement, s = Average velocity � Time

= �

�2

u v t

Substituting Equation (1),

at t� ��

( )=

2

u us

Hence s = ut + 1/2 at2 (2) Also,

� � 2

2

12

12

= � �

� �

v at at

s v at

s

(3)

From Equation (1), �

�v u

at

� ��=

2

u v v us

a


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Hence2 2�

�( )

2

v ua

s

2as = v2 � u2 Hence v2 = u2 + 2as (4) From these equations, we can derive: v = u + at s = ut + 1/2 at2 v2 = u2 + 2as Example 2.5: The driver enters a car which is parked beside the road. The driver starts the engine and then accelerates with an acceleration of 5.0m s-2. Calculate the velocity and distance travelled by the car after 7s. Solution:

Initial velocity, u = 0m s-1

Acceleration, a = 5.0m s-2

Time, t = 7s Using equation

v = u + at = 0 + (5.0 � 7) = 35m s-1 S = ut + 1/2 at2

= (0 �5) + 1/2 (5.0 � 72) = 122.5m Example 2.6: In the long jump event, Razak was running at a velocity of 3m s-1 towards the long jump pit. He needed to achieve a velocity of 6m s-1 after covering a distance of 5m before lifting himself off the ground from the jumping board. Calculate the required acceleration for Razak to be able to do so.


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Solution:

u = 3 m s-1, v = 6 m s-1; s = 5m Using equation v2 = u2 + 2as 62 = 32 + 2a(5) 10a = 36 � 9 = 27 a = 27/10 = 2.7m s-2

Free fall is an example of linear motion under constant acceleration. In this special case, a = g which is the constant accelaration due to gravity. g on earth, g ia approximately 9.8 ms-2 .

DYNAMICS

In this subtopic, we will look at three types of Newton's Laws of Motion; Newton's First Law, Newton's Second Law and Newton's Third Law.

2.3.1 Newton’s Laws of Motion

Newton's First Law states that when an object is stationary or moving with a constant velocity, it will remain as such unless an external force acts on it. This law is also commonly known as the llaw of inertia. Thus from NewtonÊs First Law: (a) An object which is at rest will remain stationary;

(b) An object which is moving with constant velocity will continue moving, unless acted upon by external forces; and

NewtonÊs Second Law states that the rate of change of momentum of an object is proportional to the resultant force F which acts on the object. The change in momentum is at the same direction as the resultant force.

2.3


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( )�dF pdt

where is p = momentum

( )�dF mvd

If mess is constant,

�

�

dvF mdt

F ma

NewtonÊs Second Law produces the equation F = ma where F = resultant force in Newton m = mass in kg a = acceleration Example 2.7: It is easier to pull a small rock compared with a big rock when you use the same force to pull the rocks (see Figure 2.8).

Figure 2.8: Example of NewtonÊs Second Law


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From F = ma, (a) When the mass of the rock is big, bigger force is needed to move or stop the

object.

(b) When the mass of the rock is increased for the same force, its acceleration decreases.

NewtonÊs Third Law states that when two objects interact, they exert equal and opposite forces on each other. If you release the air from a balloon, you will notice that the balloon will move in the opposite direction of the air that rushes out of it. The action of the air rushing out of the balloon produces an equal but opposite reaction of the balloon, thus the balloon moves up (see Figure 2.9)! This force is known as Thrust.

Figure 2.9: Example of NewtonÊs Third Law

2.3.2 Mass and Inertia

All objects have mass. The mass of an object is the quantity of matter contained in the body. A bigger mass will have a bigger inertia. A smaller mass will have a smaller inertia. What is inertia? LetÊs look at the following phenomenon:

John is a boxer. One day, he tried to push a big punching bag which was hanging stationary. John noticed that it was difficult to get the punching bag to move. When the bag finally started swinging, John then tried to stop the motion of the swinging punching bag. He noticed that it was difficult to stop the punching bag when it was in motion.


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The above phenomenon can be explained by the concept of iinertia. The swinging punching bag will continue to maintain its swing, hence the boxer feels that it is difficult to stop it. This property of matter that causes it to resist any change in its motion is known as inertia. A force is required to overcome the inertia of an object or body. The concept of inertia is related to NewtonÊs First Law of Motion. Consider the two examples shown in Figure 2.10.

Figure 2.10: Examples of inertia in our daily life

Figure 2.10(i) shows the driver of a car thrown out of the car when the car hits the wall. He was moving with the same speed of the car before the collision, so when the car hit the wall, his inertia kept his body moving at the same speed. Thus, if he fails to fasten his seat belt, he will fall forward. Figure 2.10(ii) shows a coin placed on the surface of a card on a glass. When the card is flicked, the coin falls into the glass. The inertia of the coin kept it in a stationary position, so when the card moves, it falls downwards due to gravity.

2.3.3 Momentum

The product of mass and velocity is known as mmomentum. Momentum = Mass � Velocity p = mv


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Momentum is a vvector quantity (has both direction and magnitude). The direction of the momentum follows the direction of the velocity. The right direction is denoted by the „+‰ sign while the „-„sign denotes the left direction. The SI unit of momentum is kgm s-1 or Ns. Momentum increases when: (a) The mass increases;

(b) The velocity increases; and

(c) Both of the mass and velocity increase. The study of the concept of momentum is important for predicting the motion of an object after the occurrence of a collision.

This means that the total momentum before the collision is equal to the total momentum after the collision (this means that it has been conserved) if there are no external forces acting on the system. Momentum = Mass � Velocity P = mv

2.3.4 Collision

There are two types of collisions: eelastic collision and iinelastic collision. Collisions are often classified based on what happens to the kinetic energy of the colliding objects. A collision in which the total kinetic energy is the same before and after is called eelastic. When the final kinetic energy is less than the initial kinetic energy, the collision is said to be inelastic. A stick-together collision is a perfectly inelastic collision. Collisions in one dimension are collisions that occur in a straight line. Suppose a car of mass, m1 is travelling along a road at speed u1 towards a second car of mass m2 that is moving with an initial velocity u2. What will happen when the first car hits the second car?

The PPrinciple of Conservation of Momentum states that the total linear momentum of a closed system of bodies is cconstant.


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The principle of conservation of momentum states that the total momentum in a closed system is constant or that is, total momentum before the collision is equal to the total momentum after the collision.

m1u1 + m2u2 = m1v1 + m2v2 If the collision is an elastic collision, then:

2 2 2 21 1 1 2 1 1 2 2

1 1 1 12 2 2 2

� � �M U M U M V M V

2.3.5 Projectile Motion

If you throw a ball into the air at any angle, the ball will follow a curved path (Figure 2.11a). The exact path of the ball is determined by its initial speed and angle but gravity acts on the ball to pull it downwards. This kind of motion is called the pprojectile motion, where we study the motion of an object in two dimensions. If the effects of air resistance are neglected, the path traced by the projectile will be a parabola. The shape of the curve can easily be seen by looking at water projected from a hose (Figure 2.11b).

Figure 2.11: Examples of projectile motion There are three vector parameters: displacement, velocity and acceleration, which are used to describe and analyse the motion of the object. As no vector can have any effect in a direction perpendicular to itself, the horizontal and vertical components of the parameters of the motion are completely independent of each other. Thus, to solve any problem regarding projectile motion, it is considered as two separate, independent motions in the horizontal and vertical components as if they are one-dimensional motion. We can use the equation of motion to solve problems in projectile motion.


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For aany projectile motion, its vvertical component is at constant acceleration, g while its horizontal component is at constant speed. Figure 2.12 shows two balls dropped from the height, one vertically downwards and the other ball projected horizontally outwards.

Figure 2.12: Multiple exposure photograph showing two balls falling

Source: http://www.chsdarkmatter.com

Observe that the vertical distance are the same for both balls. This distance represents the time taken for the ball to fall, indicating that an object whether projected vertically downwards or horizontally will reach the ground at the same time.

Figure 2.13: A ball rolls down a table

Source: http://francesa.phy.cmich.edu


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The ball in Figure 2.13 rolls down the table of 0.8m height with an initial velocity of 10m s-1 in the horizontal direction. It will continue to move with the constant 10m s-1 horizontal velocity (ux and vx) until it reaches the ground. As the ball was rolling on the table initially, there is no initial vertical velocity, uy. We can categorise the information given according to the two components as given in Table 2.1.

Table 2.1: Two Components of Figure 2.13

Element VVertical Component HHorizontal Component

Initial velocity, u Final velocity Acceleration, a Displacement, s Time to reach the ground, t

uy = 0 vy gravity, g = - 10m s-2

- 0.8m (downwards) t

ux = 10m s-1

vx = 10m s-1

Constant velocity, a = 0 x m from the edge of the table t

Using the equations of motion, consider the vertical motion. Use negative signs for downward movements. Using the equation sy = uyt + �at2

- 0.8 = (0 � t) + (� x � 10 � t2) 5t2 = 0.8 t = 0.4s For the horizontal motion, we can use t = 0.4s to calculate the horizontal displacements Using the equation sx = uxt + �at2 = (0.8 � 0.4) + (� � 0 � 0.42) = 0.32m


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� Scalar quantity means that the physical quantity only has magnitude.

� A physical quantity that has both magnitude and direction is called a vector quantity.

� The force obtained from the addition of two or more forces is called the resultant force.

� A single vector can be resolved into two components which is known as the resolution of vectors.

� Displacement can be defined as the change in position of the object. It is a quantity that has both magnitude and direction; a vector quantity.

Try solving this problem:

A stone thrown horizontally at a speed of 24m s-1 from the top of the cliff takes 4.0s to hit tho the sea. Calculate the height of the clifftop above the sea, and the distance from the base of the cliff to the point of impact.

ACTIVITY 2.3

1. Which image in Figure 2.6, represents a free fall? 2. Friction is on important force in our daily live. Give two examples

where fiction is important? 3. A 40 kg boy on a roller skate is standing still when he catches a 0.5

kg ball thrown at him. If the speed of the ball is 30 ms-1, how fast does he move backward?

Explain why he moves backward? 4. A football kicked by a player leaves the ground at 10 ms-1 at angle

30 above the ground. Find the range at the flight time of the ball.

SELF-CHECK 2.1


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� Speed is defined as the rate of change of distance travelled with time; a scalar quantity.

� Velocity is defined as the rate of change of displacement; a vector quantity.

� Acceleration is the rate of change of velocity; a vector quantity.

� Linear motion means that the motion occurs in a straight line.

� Motion graphs are used to describe the motion of object.

� The gradient of the displacement-time graph represents the velocity of the object.

� If the displacement-time graph is a curve graph, an increasing gradient will mean increasing velocity.

� A velocity-time graph will tell how the velocity of an object will change with time.

� The gradient of the velocity-time graph represents the acceleration of the object.

� The area under the velocity-time graph represents the distance travelled by the object.

� The four basic linear motion equations are:

(a) v = vo + at

(b) s = ut + � at2

(c) v2 = vo2 + 2as

(d) s = � (vo + v) t

where v = final velocity; vo = initial velocity; a = acceleration; t = time; and s = displacement.

� NewtonÊs Laws of Motion consist of three laws.

� Newton's First Law states that when an object is stationary or moving with a constant velocity, it will remain as such unless an external force acts on it.


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� NewtonÊs Second Law states that the rate of change of momentum of an object is proportional to the resultant force which acts on the object.

� NewtonÊs Third Law states that when two objects interact, they exert equal and opposite forces on each other.

� All objects have mass. A bigger mass will have a bigger inertia. A smaller mass will have a smaller inertia.

� The momentum of an object is defined as a vector quantity such that: Momentum, p = Mass, m �Velocity, v.

� There are two types of collisions: elastic collision and inelastic collision, which are often classified based on what happens to the kinetic energy of the colliding objects.

� A projectile is any object that has been projected at some angle into the air where the subsequent motion of the object is a curved path.

� For any projectile motion, its vertical component is at constant acceleration, g while its horizontal component is at constant speed.

Acceleration

Displacement

Elastic collision

Equation of motion

Force

Graphs of motion

Horizontal components

Inelastic collision

Kinematics

Momentum

NewtonÊs laws of motion

Projectile motion

Range

Resultant and resolution of vectors

Scalar quantity

Speed

Vector quantity

Velocity

Vertical components


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Breithaupt, J. (2000). Understanding physics for advanced level. Cheltenham:

Nelson Thornes. Cutnell, J. D., & Johnson, K. W. (1998). Physics (4th ed.). New York: John Wiley &

Sons. Giambattista, A., Richardson, B. M., & Richardson, R. C. (2004). College physics.

New York: McGraw Hill. Giancoli, D. C. (1998). Physics: Principles with applications. New Jersey: Prentice

Hall. Hartman, H. J. (2002). Tips for the science teacher. Thousand Oaks: Corwin Press. Hewitt, P. G. (1998). Conceptual physics (8th ed.). Reading: Addison-Wesley. Young, H. D., & Freedman, R. A. (2000). University physics with modern physics

(10th ed.). USA: Addison-Wesley Longman. �

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Topic 2 Forces and Motion