INTERACTIVE SIMULATION ON PROJECTILE MOTION

Lesson 3 : Gravity and Falling Bodies

GRAVITY AND FALLING BODIES

Gravity is one of the most familiar forces in nature; its effect on motion has been a subject of discussion for centuries. If an object is dropped from a great height, it can be observed that it falls with ever increasing speed until air resistance balances the effect of gravity, at which time it is said to have reached its terminal velocity. The term free falling bodies is used for objects that are moving freely under the influence of gravity, whether they are moving upward or downward. Any object that has no forces other than gravity acting on it is said to be in free fall, whether it is moving upward, downward, or in any direction.

        It is found that if air resistance can be made negligible, then falling bodies will accelerate toward the center of the earth at the same rate, regardless of their mass. The value for the acceleration of gravity, given the symbol g, has been measured on earth as  g =  9.8 m/s².  Galileo was the first to demonstrate that all bodies fall at the same rate if air resistance is negligible. ( It is often said that he did this by dropping objects of various masses from the Leaning Tower of Pisa, although there is no historical evidence that he actually used the famed tower.) Galileo’s recorded experiments settled some very old controversies about falling bodies, proving less-popular ideas to be correct.

        Even more important than his discoveries about falling objects was his breaking away from old methods of determining truth. Galileo is often credited with being the Father of Modern Science because of his forceful demonstration of the value of observation and the discoveries he made through his ingenious experiments.

        The following is a data from one of Galileo’s earliest experiments of a ball rolling down an inclined plane. His data were recorded on his notes. Galileo held a ball at the top of an inclined, grooved board and marked its position. Releasing the ball, he marked its position at the end of equal intervals of time. This is much like dropping a ball from a height, except that the effect of gravity has been “reduced” by allowing the ball to roll slowly down the inclined board rather than falling straight down. The position as measured by Galileo are given in the following table :

Time t (equal intervals)	t2	Distance  S  (points)	S/ t2
1	1	33	33.0
2	4	130	32.5
3	9	298	33.1
4	16	526	32.9
5	25	824	33.0
6	36	1192	33.1
7	49	1620	33.1
8	64	2104	32.9

       The observations show what was already known quantitatively to Galileo and others of his time – that a rolling (or falling) object picks up speed as it continues to roll (or fall). However, the debt we owe to Galileo is for his careful measurements and his quantitative (mathematical) interpretation of the data. His object was to find a general rule describing how distances increase with increasing time of fall. After some trial and error, and with considerable insight, Galileo realized that the distance traveled was proportional to the square of the elapsed time.

                                                        S at²  ==>   S = h = ½ at²    

 Problems  

       1. A ball is thrown vertically up with an initial velocity of 15 m/s. How high does the ball rise from its projection point ? How long does it take for this ball to reach the highest point. How high does it go in 2 seconds ? in 3 seconds ?  What is the time required to travel a height of  9 m ?  5 m ?   

          Ans. ( 11.48 m ,  1.53 s , 10.4 m ,  0.9 m , 0.82 s ,  2.24 s ,  0.38 s ,  2.68 s )

       2. A rock is dropped from a bridge 55 m high relative to the water of a river below. How long will it take for the rock to reach the surface of the water ? Calculate the positions of  the rock  0.5s,  1.25 s, 2s after it was release relative to the water.  

       3. A metal sphere is dropped from a 50 m high tower. Determine the height traveled by the  sphere in the time interval from 0.25 s  to  1.25 s. 

     4. "Khalifa Tower", pronounced in English (/ˈbɜrdʒ kəˈliːfə/), known as Burj Dubai before its inauguration, is a skyscraper in Dubai, United Arab Emirates. It is the tallest man-made structure in the world, standing at 829.8 m (2,722 ft).  If an object falls from the top of this tower,  how many seconds does it reach the ground ?  What is the height travelled by the object in the interval  between  6 seconds and 10 seconds after falling?

Lesson 2 : Motion

Motion

      Motion is apparent in widely ranging phenomena, from blood cells squeezing through capillaries to planets moving across the sky. Motion is the displacement of an object with respect to objects that are at rest. Historically, motion was one of the first phenomena to be studied carefully. Some progress was made in the understanding of motion in ancient times, particularly by the philosophers of classical Greece, but it was not until the Renaissance that the basic laws of motion were discovered. Many individuals made important contributions, but two stand above the rest : Galileo Galilei ( 1564 – 1642 ) and Isaac Newton ( 1642 – 1727 ).  If Galileo’s predecessors had placed a greater value on experimentation, they might have made more progress than they did. Instead most natural philosophy was based on logical argument and the constraining  influence of a particular school of thought. The transition that Galileo and others made from dogma to experimentation was not without pain; Galileo himself was forced by the Inquisition to recant his work and lived the last years of his life under a form of house arrest.

      The central ideas regarding motion developed by Galileo and Newton remained essentially intact until 1905, when Albert Einstein ( 1879 – 1955 ) published his paper on the theory of relativity. Even today, the classical theory of Galileo, Newton and others describes motion with extremely good precision as long as the object  being described moves slower than about 1% of the speed of light. The study of motion is kinematics, motion being the displacement of objects with respect to objects that are at rest. Kinematics comes from the Greek word kinema, meaning motion, the same root from which we get the word cinema. Kinematics describes the position and motion of objects in space as a function of time but does not consider the causes of motion.( It deals with motion without considering the forces causing the motion ). The study of the causes of motion is dynamics which relates motion to the forces causing it and to the properties of the moving system.

      Kinematics provides the means for describing the motions of varied things as planets, golf balls, and subatomic particles. Because of its precision and generality, mathematics is the natural language for kinematics. To adequately describe motion, one must be able to say where something is located within a given reference frame. Reference frame is a physical entity, such as ground, a room or a moving car, to which we refer the position and motion of the objects.

      To say that space is three dimensional, it means that three numbers are needed to completely locate the position the position of an object. A system for assigning these 3 numbers, or coordinates, to the location of a point in a reference frame is called coordinate system. Because the coordinate system is a mathematical construction, you are free to choose the system that you want, orient it as you wish, and place its origin wherever you prefer.       

  TIME, DISPLACEMENT, VELOCITY AND ACCELERATION

            Time is measured in terms of change. If nothing changes, then it is impossible to tell that time has passed. All devices that measure time measure change; i.e., days are measured are measured by the change in position of the sun in the sky, clocks measure elapsed time by the change in position of their hands.

            Displacement  is the location of an object relative to a reference point. Displacement is specified by the distance from a reference point (magnitude) and the direction to get to the present location. This implies that displacement is a vector quantity which has magnitude and direction. Distance has no given direction and has only magnitude. It is a scalar quantity.

            Velocity and Speed. Speed is time rate of change of position while velocity is time rate of change of displacement. Velocity can also be describe as speed in a specific direction.

                                                ν =   Change in displacement     =    Δ S     ,  ν   is average velocity 

                                                            Change in time                    Δ t

            Acceleration is the time rate of change of velocity or the speeding up or the slowing down of bodies in motion. Acceleration is a vector quantity and has both magnitude and direction.

a)      Positive acceleration ( acceleration,) –  the speeding up of bodies in motion ( ν_f  >  ν₀ ).

b)      Negative acceleration ( deceleration) –  the slowing down of bodies in motion ( ν_f  <  ν₀ ).

        Average acceleration, a  =    Change in velocity    =    Δ ν      
                                                       Change in time                Δ t 

  

Making and Recording Measurements

MAKING AND RECORDING MEASUREMENTS

Reasons for uncertainty in Measurements

1.      The limitations inherent in the measuring instrument.

2.      The conditions under which the measurement was made.

3.      The different ways under which the person uses or reads the measuring instrument.

Terminologies:

1. Fundamental or base unit – the standard unit for length, mass and time.
2. Derived unit a combination of any of the three fundamental base units; i. e.  m/s, m/s²,  ft², m³ etc.
3. Accuracy – refers to the closeness of a measurement to the standard value for a specific physical quantity. It is express either as an absolute error or relative error.
4. Absolute error ( E_A ) is the actual difference between the observed ( O ) or measured value and the accepted value ( A ).

                                                      E_A = | O – A |

5. Relative error ( E_R ) I expressed as a percentage error and is often called a percentage error.

                                                      E_R = E_A/A

6. Absolute deviation ( DA ) is the difference between a single measured value ( O ) and the average         ( M ) of several measurements made in the same way.

                                                      DA = | O – M |

7. Relative deviation ( D_R ) is the percentage average deviation of a set of measurements.

                                                      D_R = D_A/M

8. Precision is the agreement among several measurements tat have made in the same way. It tells how much reproducible the measurements are and is express in terms of the deviation.
9. Tolerance is the degree of precision obtainable in a measuring instrument.
10. Significant figure are those digits in a number that are known with certainty plus the digit that is uncertain.

Lesson 1

Physics :  Introduction

            Physics is a major science, dealing with the systematic study of the basic properties of the universe, the forces they exert on one another, and the results produced by these forces. Physics is closely related to the other natural sciences and, in a sense, encompasses them. Chemistry, for example deals with the interaction of atoms to form molecules. Much of modern geology is largely a study of the physics of the earth and is known as geophysics. Astronomy deals with the physics of the stars and outer space. Even living systems are made up of fundamental particles and, as studied in biophysics and biochemistry, they follow the same type of laws as the simpler particles traditionally studied by a physicist.

The emphasis on the interaction between particles in modern physics, known as the microscopic approach, must often be supplemented by a macroscopic approach that deals with larger elements or systems of particles. This macroscopic approach is indispensable to the application of physics to much of modern technology. Thermodynamics, a branch of physics developed in the 19^th century, deals with the elucidation and measurement of properties of a system as a whole and remains useful in other fields of physics; it also forms the basis of much of chemical and mechanical engineering. Such properties as the temperature, pressure and volume of a gas have no meaning for an individual atom or molecule; these thermodynamic concepts can only be applied directly to a very large system of such particles. A bridge exists, however, between the microscopic and macroscopic approach; another branch of physics; known as statistical mechanics, indicates how pressure and temperature can be related to the motion of atoms and molecules on a statistical basis.  

Physics emerged as a separate science only in the early 19^th century, until that time a physicist was often also a mathematician, philosopher, chemist, biologist, engineer, or even primarily a political leader or an artist. Today, the field has grown to such an extent that with few exceptions modern physicists have to limit their attention to one or two branches of the science. Once the fundamental aspects of a new field are discovered and understood, they become the domain of engineers and other applied scientist. The 19^th century discoveries in electricity and magnetism, for example, are now the concentrations of electrical and communication engineers; the properties of matter discovered at the beginning of the 20^th century have been applied in electronics; and the discoveries of nuclear physics, have passed into the hands of nuclear engineers for applications to peaceful or military uses.

MATHEMATICS as a language of science

     Mathematics is the language of physics; that is when ideas in science are expressed in mathematical terms:         

                   1. They are unambiguous.    

                   2. They do not have double meanings, that so often confuse the discussion of ideas expressed in

                        common language.

                   3. They are easier to verify or disprove by experiment.

                   4. The methods of mathematics and experimentation led to enormous success in science.

                   5. The abstract mathematics developed by mathematicians is often years later found to be the

                         exact language by which nature can be described.

    Mathematics is the language of physics does not mean that mathematics is physics or physics is mathematics.

THE SCIENTIFIC METHOD – is a method that is extremely effective in gaining, organizing, and applying new knowledge. The steps are :

1.      Recognize a problem.

2.      Make an educated guess --- a hypothesis. Hypothesis is an educate guess that is only considered factual after it has been demonstrated by experiments. If a hypothesis has been tested over and over again and has not been contradicted it may become known as a law or principle.  

3.      Predict the consequences of the hypothesis

4.      Perform experiments to test predictions.

5.      Formulate the simplest general rule that organizes the three main ingredients --- hypothesis, prediction, and experimental outcome --- into a theory.

The success of science has more to do with an attitude common to scientists than with a particular method. This attitude is one of inquiry, observation, experimentation and humility.

THE DOMAIN OF PHYSICS

A.  According to size of objects studied

     1.  Quantum domain – the domain of small objects. Objects are considered small if their sizes are

            comparable to or smaller than the size of an atom.

    2.   Non-quantum domain – the domain of large objects. Objects are considered large if they are 
           larger than the size of an atom.

B.  According to speed of objects studied

     1.  Relativistic domain – the domain at high speed, that is if the speed of the moving object is comparable               to the speed of light.

     2. Non-relativistic domain – the domain at low speed, that is the speed of the moving object is less 
              than the speed of light.

C.  Newtonian domain – a combination of the division according to size and speed. It is the domain of large

            objects at low speeds, the one we deal in our daily lives. (In honor of Sir Isaac Newton, the 
            17^th century physicist who played the key role in developing the physics of large objects moving at  
            low speed).

D.  Mechanics – is the study of the relation between the force and the resulting motion. It seeks to account  

               quantitatively for the motion of objects having given properties in terms of the force acting on them.

      1.  Newtonian mechanics – is the mechanics of the Newtonian domain. It deals with systems containing    

            objects which are large and which move at low speed.

2. Relativistic mechanics – is the mechanics of the relativistic domain. In 1905, Einstein showed that a        different approach was necessary for the study of objects moving at speeds so high as to  be                 comparable   to the speed of light. 
3. Quantum mechanics – is the mechanics of the quantum domain. It was developed about the same time with relativistic mechanics by Max Planck, Louis de Broglie, Erwin Schrodinger and others. They found out that the Newtonian mechanics could not explain the motion of objects whose size is in the atomic scale or smaller.

 E.   Electromagnetism – is the study of the properties and consequences of the electromagnetic force, which 

              is one of the fundamental forces in nature. The fundamental forces are gravitational force, 

              electromagnetic force, strong nuclear force and weak nuclear force.

 F.   Solid-state physics is a branch of physics that deals with the properties of solids. A particular problem 

            in solid – state physics, for instance the properties of materials use in transistors, is solve by 

            employing the mechanics of whichever domain is most appropriate.  

 G.  Heat  and Thermodynamics

   THE FUNDAMENTAL MEASURABLE QUANTITIES IN PHYSICS

    1.  Length               3.  Time                       5.  Luminous intensity             7.  Molecular quantity

    2.  Mass                 4.  Temperature           6.  Electric charge ( current )

   THE FUNDAMENTAL MEASURABLE QUANTITIES IN MECHANICS

               1.  Length                 2.  Mass                       3.  Time

  Measurement is a scientific comparison between an unknown quantity to a fixed known quantity called standard.

 Systems of Measurement

1.      English system  (British Engineering system) – originated in England

2.      Metric system – originated in France

  Systeme International d’Unites ( SI ) adopted by  the International Bureau of Weights and Measures in 1960. 

   The units of the MKS is adopted as the base units of the SI system.

Base Units of each System of measurement

Measurable Quantities in Mechanics	Metric System		English System
	CGS	MKS	FPS
Length	Centimeter ( cm )	Meter ( m )	Foot ( ft )
Mass	Gram ( g )	Kilogram (kg )	Slug ( lbm )
Time	Second ( s )	Second ( s )	Second ( s )

Reasons for adopting the Metric system:

1.      It is scientifically planned.

2.      It is a decimal system.

3.      It is universally accepted.

 DISADVANTAGES OF THE ENGLISH SYSTEM

            1 yard = ( King Henry I ) distance from the tip of his nose to the end of his thumb

            1 inch ( 1324 ) = length of three grains of barleycorns laid end to end

            1 mile = 1000 double step of an average soldier

            1 foot = length of the foot of the king