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.