Physics Gr12 PDF

First Published August 2023 by the Federal Democratic Republic of Ethiopia, Min­istry of Education, under the General Education Quality Improvement Program for Equity (GEQIP-E) supported by the World Bank, UK's Department for Inter­national Development/DFID-now merged with the Foreign, Common wea...

First Published August 2023 by the Federal Democratic Republic of Ethiopia, Min­istry of Education, under the General Education Quality Improvement Program for Equity (GEQIP-E) supported by the World Bank, UK's Department for Inter­national Development/DFID-now merged with the Foreign, Common wealth and Development Office/FCDO, Finland Ministry for Foreign Affairs, the Royal Norwegian Embassy, United Nations Children尹s Fund/UNICEF), the Global Partnership for Education (GPE), and Danish Ministry of Foreign Affairs, through a Multi Donor Trust Fund. ©2023 by the Federal Democratic Republic of Ethiopia, Ministry of Education. All rights reserved. The moral rights of the author have been asserted. No part of this textbook reproduced, copied in a retrieval system or transmitted in any form or by any means including electronic, mechanical, magnetic, photocopying, recording or otherwise, without the prior written permission of the Ministry of Education or licensing in accordance with the Federal Democratic Republic of Ethiopia as expressed in the Federal Negarit Gazeta, Proclamation No. 410/2004 -Copyright and Neighboring Rights Protection. The Ministry of Education wishes to thank the many individuals, groups and other bodies involved鈤湣directly or indirectly - in publishing this Textbook. Special thanks are due to Hawassa University for their huge contribution in the development of this textbook in collaboration with Addis Ababa University, Bahir Dar University and Jimma University. Copyrighted materials used by permission of their owners. If you are the owner of copyrighted material not cited or improperly cited, please contact the Ministry of Education, Head Office, Arat Kilo, (P.O.Box 1367), Addis Ababa Ethiopia. 978-99990-0-035-2 Contents 1 Application of physics in other fields 1 1.1 Physics and other sciences....................... 2 1.2 Physics and engineering........................ 9 1.3 Medical physics............................. 12 1.4 Physics and defense technology.................... 17 1.5 Physics in communication....................... 20 2 Two-dimensional motion 24 2.1 Projectile motion............................ 25 2.2 Rotational Motion............................ 39 2.3 Rotational Dynamics.......................... 50 2.4 Planetary motion and Kepler’s laws.................. 54 2.5 Newton’s law of universal Gravitation................ 59 3 Fluid Mechanics 69 3.1 Fluid Statics................................ 69 3.2 Pressure in fluids at rest......................... 82 3.3 Archimedes’ principle.......................... 96 3.4 Fluid flow................................. 103 3.5 Safety and high pressure........................ 107 4 Electromagnetism 117 4.1 Magnets and Magnetic field...................... 118 4.2 Magnetic field lines........................... 120 4.3 Current and Magnetism........................ 122 4.4 Electromagnetic Induction....................... 126 4.5 Faraday’s Law of electromagnetic Induction............. 129 4.6 Transformers............................... 131 4.7 Application and safety......................... 134 5 Basics of electronics 142 5.1 Semiconductors............................. 143 5.2 Diodes and their Functions...................... 147 5.3 Rectification............................... 150 5.4 Transistors and their application................... 154 i ii CONTENTS 5.5 Integrated Circuits............................ 161 5.6 Logic gates and logic circuits..................... 163 5.7 Application of electronics....................... 170 Index 177 Unit 1 Application of physics in other fields Introduction Brainstorming Science is a collection of different scientific fields, or disciplines. It is the union of question1.1 these fields that helps us to understand the world in which we live. Although there (i) List as many are many scientific disciplines with a special character and history of their own, other sciences or each discipline is dependent on and reinforces the other disciplines. Scientific specializations as disciplines do not have fixed borders. Each discipline benefits from advances in possible in which other areas of science. Understanding the contribution that a scientific discipline the word physics can make to others is important for the collaborative development all scientific appears in their disciplines, and their contribution to society and the environment. names. (ii) How is At the end of this unit, you will be able to: knowledge of Comprehend the contribution of physics to the betterment of society. physics used or applied in Understand the relation of physics to other sciences. these sciences or specializations? Recognize and appreciate the place of physics in advancement of Technology. Appreciate the contribution of Technology to the advancement of Physics. Advocate physics as an important field of study to address societal issues and challenges. 1 2 Unit 1 Application of physics in other fields 1.1 Physics and other sciences Physics is the most essential field of science and it has a strong influence on most scientific developments. We can find different concepts of physics in many of the Brainstorming modern sciences. question 1.2 (i) Identify some At the end of this section, you will be able to: relationships explain the relationship of physics with chemistry, biology, geology and between physics astronomy. and chemistry. (ii) What physics concept is applied Physics and chemistry in separation of dissolved salt Physics and chemistry may overlap when the system under study involves matter from water by composed of electrons and nuclei. Fundamental laws that govern the behavior of evaporation matter apply to both chemistry and physics. Both physics and chemistry are method? concerned with matter and its interaction with energy. The theory and various rules about atoms which are important in chemistry are ultimately explained in principle by atomic and subatomic particle physics. Chemists and physicists use Discussion similar method to study the interaction of large number of particles which have question 1.1 complicated interaction beyond the capacity of any computer, and the capacity of the human mind. tDear students, chemistry studies minute entities The physics of atoms and subatomic particles is critically important for like elements, understanding how individual atoms are joined by covalent bonds to form atoms, molecules molecules. The physics of atoms and subatomic particles can also provide and ions which quantitative insight into ionic and covalent bonding processes by explicitly cannot be seen with showing which molecules are energetically favorable to which others and the your naked eyes. magnitudes of the energies involved. How do chemists collect information, like composition, The physics of energy related to heat tells chemists whether a particular reaction structure, properties is energetically possible in the direction in which it is written, and it gives the and the changes composition of the reaction system at equilibrium. The physics of heat energy they undergo during provides a bridge between the macroscopic properties of a substance and the a reaction with individual properties of its constituent molecules and atoms. other substances, from these minute invisible particles? Spectroscopy is the study of the interaction between matter and electromagnetic radiation as a function of the wavelength or frequency of the radiation. 1.1 Physics and other sciences 3 Spectroscopy is a fundamental examining tool in many fields including physics and chemistry. Most of what we know about the structure of atoms and molecules comes from spectroscopic study. The spectroscopic techniques are developed by collaborative work of physicists and chemists. In general, the study of matter and electricity in physics is fundamental towards the understanding of concepts in chemistry, such as atomic structure, molecular structure, X-ray diffractions, radioactivity, periodic properties of elements, nature of valency, chemical bonds in molecules, crystal structure of solids and others. This shows that chemistry is rooted in atomic and molecular physics. Physics and biology To understand how life works, it is essential to understand physics. Physics can Brainstorming explain the human body like the mechanics of human motion, the energetics of question 1.3 metabolism, the fluid dynamics of blood flow through vessels, the mechanisms List some physics for speaking and hearing, and the optical imaging system we call the eye. In this concepts in our section some of the physics concepts within a living things are briefly explained. body. Physics of Newtonian mechanics and biology The Newtonian mechanics tells us how different animals and their body parts move. The combination of knowledge of physics and biology help to explain how athletes run fast and why the fastest animal in the world, cheetah, runs fast. Newtonian mechanics tells us that a body is in stable equilibrium under the action of gravity if its center of mass is directly over its base of support. Under this condition, the reaction force at the base of support cancels the force of gravity and the torque produced by it. If the center of mass is outside the base, the torque produced by the weight tends to fall the body. A person falls when his center of gravity is displaced beyond the position of the feet. The wider the base Figure 1.1 (a) Center of gravity on which the body rests and the closer the center of gravity of it to the base, the of a freely standing person (b) A more stable it is; that is, the more difficult it is to fall it. How we can use our limbs person carrying unbalanced load to do different jobs can also be explained by Newtonian mechanics. stands bending 4 Unit 1 Application of physics in other fields Discussion question 1.2 (i) Why it is difficult to stand on one foot than the two feet? Why we are more stable when we sleep than we are standing? (ii) Figure 1.1 (a) shows freely standing person and Figure 1.1 (b) shows the way a person carrying a load stands. Why the person carrying the load stands bending in Figure 1.1(b) than the person in Figure 1.1(a)? Physics of fluid flow and biology Discussion The physics of fluid flow, like viscosity, equation of continuity and turbulent flow, question 1.3 is very important in understanding the circulation of blood and blood pressure in (i) Have you the multicellular organisms. Soft-bodied animals (such as the sea anemone and observed when the the earthworm) that lack a firm utilize Pascal’s principle to produce body motion. earthworm moves? How do earthworms locomote without limbs? (ii) How a blood moves against gravity from the feet to the heart or from the heart to the brain when a person stands erect? Figure 1.2 Sound wave generating organ in human being. Activity 1.1 Physics of sound wave and biology Dear students, Sound is a mechanical wave produced by vibrating bodies. The vocal cords please check produce sound when they come together and then vibrate as air passes through whether inhalation them during exhalation of air from the lungs. This vibration produces the sound or exhalation of air enables formation wave for your voice. When the human vocal cords (Figure 1.2) set into vibrational of a controlled motion, the surrounding air molecules are disturbed and are forced to follow sound. the motion of the vibrating body. The vibrating molecules in turn transfer their motion to adjacent molecules causing the vibrational disturbance to propagate away from the source. When the air vibrations reach the ear, they cause the 1.1 Physics and other sciences 5 eardrum to vibrate; this produces nerve impulses that are interpreted by the brain. Physics of electricity and biology Many life processes involve electrical phenomena. The nervous system of animals and the control of muscle movement, for example, are both governed by electrical interactions. The very important electrical phenomena in living organisms are found in the nervous system of animals. Specialized cells called neurons form a complex network within the body which receives, processes, and transmits information from one part of the body to another. The center of this network is located in the brain, which has the ability to store and analyze information. Based on this information, the nervous system controls various parts of the body. The messages are electrical pulses transmitted by the neurons. When a neuron receives an appropriate stimulus, it produces electrical pulses that are propagated along its cablelike structure. Optical physics and biology Light is the electromagnetic radiation in the wavelength region between about 400 nm and 700 nm. Although light is only a tiny part of the electromagnetic spectrum, it is very important in both physics and biology. Light has fundamental roles in living system because of its paramount importance. Most of the electromagnetic radiation from the sun that reaches the Earth’s surface is in this region of the spectrum, and life has evolved to utilize it. In photosynthesis, plants use light to Discussion convert carbon dioxide and water into organic materials, which are the building question 1.4 blocks of living organisms. Animals have evolved light-sensitive organs, like Discuss how our the eyes, which are their main source of information about the surroundings. eyes use light to see Some bacteria and insects can even produce light through chemical reactions. objects? Optical physics, which is the study of light, includes topics such as microscopes, telescopes, vision, color, pigments, illumination, spectroscopy, and lasers, all of which have applications in the life sciences. Physics and astronomy Astrophysics is the study of the physics of heavenly objects, called astronomical objects, in the sky like the solar system and its constituents, the properties, birth, life and death of stars, interstellar gas and dust, galaxies and clusters of galaxies, 6 Unit 1 Application of physics in other fields and finally the study of the Universe as a whole. Brain storming question 1.4 Newton’s laws of motion and astronomy Most of the Newton’s law of gravitation is used to describe the motion of the moon around astronomical a planet and the motion of the planets around the sun. Newton was able to objects are far explain why Kepler’s Laws described planetary motion using his laws of motion from the earth where astronomers and gravity. The knowledge of centripetal force and centrifugal force from our live. How can physics knowledge greatly help as to understand what keeps objects in orbit astronomers around others. This applies to planets orbiting the Sun, moons orbiting planets, get information and artificial satellites in Earth0 s orbit. about these far objects? What Physics of electromagnetic wave and astronomy physics concepts do astronomers use to Astronomers collect information about the radiation from space objects to study study astronomical the birth and death of stars, how hot objects are, how far away they are, even how objects? the universe was formed. Astronomers use telescopes that detect different parts of the electromagnetic spectrum. Each type of telescope can only detect one part Brainstorming of the electromagnetic spectrum. There are radio telescopes, infrared telescopes, question 1.5 optical (visible light) telescopes and so on. We can’t see most of the radiation What information detected, so computers turn data into images we can see. The colour we observe do you expect on these image are called false colours because computers have taken the data from studying the from wavelengths we can’t see and presented them as colours that can be seen as electromagnetic shown in Figure 1.4. wave which comes from astronomical objects? Figure 1.4 Detection of different components of electromagnetic radiation Figure 1.3 Entoto space by telescope observatory telescope Astronomers use light to measure distance of astronomical objects. For more 1.1 Physics and other sciences 7 distant objects, we can measure distances by using brightness of objects, since objects will appear fainter if they are at larger distances than identical nearby object. Measuring the apparent brightness of an object gives its distance if we know its true brightness. true brightness apparent brightness ∝ distance 2 This is known as the inverse square law of apparent brightness. The true Exercise 1.1 brightness is also known as the luminosity. Astronomers also used light year as What is the distance an alternate astronomical distance measuring unit. A light year is the distance that light can that light travels in one year. travel in one year in kilometer? Physics of atoms and astronomy Atomic astrophysics is concerned with performing atomic physics calculations Brain storming that will be useful to astronomers and using atomic data to interpret question 1.6 astronomical observations. The astronomers’ only information about a What do emission particular object comes through the light that it emits, and this light arises and absorption of through atomic transitions. light by an object tell us? The physics that explains emission and absorption of radiation is closely related to the structure and energies of individual atoms that form the astronomical objects. When electrons of atoms jump from higher to a lower atomic orbit, photon is emitted. Photon is absorbed if electron jumps from lower electron orbit to higher electron orbit (Figure 1.5). The emission and absorption of radiation depends on the characteristics of individual atoms, and helps to measure something about the compositions, temperatures, and motions of stars by studying their spectra. You can visit the link: https://phet.colorado.edu/en/simulations/hydrogen-atom to observe the Bohr atomic model for hydrogen atom. In the interstellar matter, atoms are heated by nearby stars. This results in knocking of electrons to higher energy orbits. These electrons fall down to lower energy orbits emitting light of precisely the wavelength that corresponds to the energy change between the two orbits. The nature of the emitted light depends on the temperature. Thus, emitted light can be used to determine both compositions and temperatures of astronomical objects. 8 Unit 1 Application of physics in other fields Figure 1.5 The Bohr atomic structure and electron transitions Physics and geology The study of different parts of Earth is called Earth science. Earth Science deals with all aspects of the Earth including molten lava, icy mountain peaks, steep valleys and towering waterfalls, the atmosphere high above the earth as well as the Earth’s core far beneath the surface. Geology is a branch of Earth science that studies the solid and liquid matter that makes up Earth and the different Figure 1.6 Mineralogists focus on processes on these matters. all kinds of minerals. The understanding of geological processes demand the understanding of the Activity 1.2 different concepts of physics like force, optics, atomic structure, electromagnetic radiation, heat and heat flow, electricity and magnetism, stress and strain, waves Dear students, please organize including sound wave and fluid flow. In geology, these physics concepts are used yourself in five to study the following properties of rocks and minerals: electrical properties, groups to visit density, magnetization, radioactivity, elasticity and more. To study these your other science properties of rocks and minerals,the geologist take samples from different layers teachers like of the earth through excavation. They can also study different properties of rocks chemistry, biology, and minerals by sending different types of waves from the surface of the earth geography, history without excavation. and ICT to ask them how the knowledge Geology has so many branches that most geologists become specialists in one of physics is used in area. For example, a mineralogist studies the composition and structure of their disciples. Your teacher will help minerals such as halite (rock salt), quartz, calcite, and magnetite (Figure1.6). you how to proceed Geological knowledge is also used to detect or infer the presence and position of in your activities. 1.2 Physics and engineering 9 economically useful geological deposits, such as ore minerals, fossil fuels and other hydrocarbons, geothermal reservoirs, and groundwater reservoirs. Brainstorming 1.2 Physics and engineering question 1.7 (i) List importance At the end of this section, you will be able to: of physics in Relate the Newtonian mechanics with civil engineering engineering. (ii) Should we learn List different concepts of physics used in mechanical engineering physics before we learn engineering? Relate electromagnetism to electrical and electronics engineering Explain how technology contributes to the development of physics Physics generates fundamental knowledge that can be used by different branches of engineering. Physical concepts, such as Mechanics, Thermodynamics, Electromagnetism, Atomic Physics, Molecular Physics, Optics, Nuclear Physics etc., are important knowledge inputs in different engineering branches. Engineering branches such as civil, mechanical, electrical, etc., are basically governed by physical laws. It is difficult to solve many of the complex engineering problems without understanding the physics behind it. If one understands the laws of nature using physics, then one can use that knowledge to predict what will happen to the things one builds in engineering. Figure 1.8 Civil Engineering in (a) Suspension Bridge (Abay Bridge) (b)Building construction (Africa Union Head Quarter, Addis Ababa) (c) Transportation Engineering (Gotera interchange road) Figure 1.7 Civil Engineering in Civil engineering ancient civilization(a) Axumite Obelisks (b) Lalibela Betegiorgis Civil engineering concerns designing and building skyscrapers, roads,bridges, rock hewen church (c) Harar Jugol (The Arthur Rimbaud Cultural dams, and railways using our physics knowledge of forces, fluid pressure, gravity, center) (d) Egyptian Pyramids 10 Unit 1 Application of physics in other fields and others. Civil engineering has been known since ancient civilizations in Ethiopia, Egypt and others (Figure1.7), where large buildings were built such as temples, pyramids and palaces with engineering designs. However, the major change in civil engineering resulted from the development of physics particularly after the development of laws of motion, power and energy in the eighteenth century. Advance in accuracy of measurements and calculations in Civil engineering results in construction of complex sky scraper buildings, transportation/traffic systems/ engineering, suspension bridges and others in worldwide. Mechanical engineering Mechanical engineering uses knowledge of mathematics, science mostly physics and materials science to create mechanical systems like engines, manufacturing equipment and vehicles. The physics concepts like mechanics, dynamics, thermodynamics, forces, stresses and aerodynamics are mostly used in mechanical engineering in dealing with aircraft, watercraft, engines, robotics, weapons, cars, hydraulics and others. A mechanical engineer takes one or more of these concepts to create a mechanical system that operate without failure. Figure 1.10 Mechanical engineering design products Electrical engineering Electrical engineering involves designing electrical circuits including motors, electronic appliances, optical fiber networks, computers, and communication links. Electrical engineers often need to convert electrical energy to other forms Figure 1.9 Areas electrical of energy like mechanical and thermal energy. Therefore electrical engineering engineering concerns (a)electrical demands the understanding of some physics concepts like electromagnetism, system for installation and mechanics, thermodynamics and others. maintainance (b) Electric power transmission and management (c) Electric circuit 1.2 Physics and engineering 11 Chemical engineering Chemical engineering involves the production of products through chemical processes. This includes designing equipment, systems, and processes for refining raw materials and for mixing, compounding, and processing chemicals. The laws of physical chemistry and physics govern the practicability and efficiency of chemical engineering operations. Particularly, chemical engineering requires an understanding of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics which are dealt by molecular physics. Furthermore, concept of energy changes, deriving from thermodynamic considerations, are very important in chemical engineering. Figure 1.11 Some chemical Engineering products (a) plastics (b) petroleum products (c) detergents and (d)paints Brainstorming Technology generating new physics question 1.8 There is a fundamental connection between physics and technology. Without What is the knowledge of physics, most of the technologies we know well today could not be contribution of available for the society. Dear students, do you think the revere also is true i.e., technology in the can technology give rise to new physics? development of physics? Science and technology are two things that are completely interwoven. Science seeks to understand the natural world using technology. Engineering uses scientific discoveries to design products and processes to solve the societal 12 Unit 1 Application of physics in other fields problems. This products and processes are what we call technology which are important for the scientists as well as for the engineers. Technology wouldn’t exist without science, and science wouldn’t be as effective without technology. The technology that was developed using science can help to do even more science. Many modern scientific experiments wouldn’t be possible without technology. The rocket technology allowing blasting off from the earth enables us to take scientific measurements in space. The discovery of X-ray technology helped in further development of physics. Technology helped in study of atomic structure, spectral analysis, etc. The relation of science/physics, Engineering and technology can be schematized as shown in Figure 1.12. Activity 1.3 Dear students, Carefully think about your living area including your school. (i) What technologies and engineering products are available? Explain Figure 1.12 Interrelation of how physics is used in these technologies and engineering products. physics, engineering and technology (ii) Try to find also the latest technology and engineering products based on physics discoveries. You can use sources like internet, library, expert visit and media. Your work has to be presented in group. (iii) Present a poster on the title “Contribution of different technologies on the advancement of physics”. Brainstorming question 1.9 List as many 1.3 Medical physics medical diagnostic instruments you At the end of this section, you will be able to: know. Try to explain Explain the relation of physics with medicine what information do these instruments Explain how the magnetic property of water is used in MRI for diagnosis collect from the body part being Differentiate the working principle of conventional x-ray and computer diagnosed? tomography (CT) scan Discuss how sound wave is used for diagnosis Explain how radiation is used for cancer and tumor treatment Medical physics is a branch of physics that deals the applications of principles of 1.3 Medical physics 13 physics to medical diagnosing and treating abnormal tissues. The discovery of X-rays by Wilhelm Conrad Roentgen in 1895 brought a revolution in the fields of science and medicine and it has opened a path to a new interdisciplinary branch, medical physics. The first X-ray photograph was made by Roentgen himself in late 1895 (Figure 1.13). Medical imaging refers to several different technologies that are used to view the human body in order to diagnose, monitor, or treat medical conditions. There are several imaging techniques that can provide imaging of biological samples. Some techniques used for imaging are electromagnetic (optical, X-ray, magnetic resonance imaging (MRI), thermography); other techniques are acoustic (ultrasound), chemical, and electrical. Among the most prolific ones are MRI, X-ray computed tomography (CT scan), and high-frequency ultrasound. Figure 1.13 Wilhelm Conrad Magnetic Resonance Imaging (MRI) Roentgen0 s hand first x-ray image Magnetic resonance is absorption or emission of electromagnetic radiation by electrons or atomic nuclei in response to the application of certain magnetic fields. MRI uses the physical principle of magnetic resonance that was first described by Felix Bloch and Edward Purcell in 1946. Paul Lauterbur and Peter Mansfield described how to acquire MR images from the human body. How is MRI working?: Getting an MRI image depends upon the presence of protons in the body. Protons are free hydrogen atom (proton without electrons). They are abundant in the body as most part of our body consists of water which contains two hydrogen and one oxygen atoms. With their net positive charge, Figure 1.14 An MRI image of the protons are small magnets each having a north and a south pole. brain. The MRI technique yields detailed visualization of soft tissue structures with a resolution of How do we get an image from these oscillation of flipped hydrogen protons in the about 0.5 mm. brain? MRI uses the movement of these small magnets within a magnetic field to generate an image. Within the constant magnetic field of an MRI scanner, these Discussion small magnets arrange themselves parallel to constant magnetic field. When a question 1.5 current pulse (a current that flows for very short period of time) is applied to How can magnetic the patient’s tissue, this parallel arrangement of the small magnets is disturbed. vibration be used to When the current pulse is off the small magnets back to their parallel arrangement form MRI image of releasing energy that they absorbed from the pulse. Different tissues in the body the body part and used for diagnosis? give off different amounts of energy. A special device detects the released energy as an electrical current. The electrical current is transformed in to an image via a 14 Unit 1 Application of physics in other fields computer. Because protons in the different kinds of tissues in the brain, such as gray matter, white matter and blood, all give off different amounts of energy, the result of the transformed energy is a highly detailed image of the tissue inside the brain. X-Ray computerized tomography (CT) scan Brainstorming question 1.10 An X-ray imaging is based on the absorption of X-rays as they pass through the different parts of a patient’s body. Differences in the densities of body tissue allow Dear student, share us to see inside the body by creating a shadow gram. The body is composed of your X-ray and CT scan diagnostic tissues containing many different elements, which vary by atomic number (the experience. number of protons in the nucleus). The higher the atomic number, the denser the element and the more effectively the X-ray is blocked. For example, when X-ray strikes the calcium in cortical bone, it is blocked, and on the radiographic image the bone will appear white. When an X-ray strikes less dense element like nitrogen, it passes all the way through. Therefore, the air-containing lung will appear darker, approaching black on the radiographic image. When a fracture extends through the bone, the fracture line will be dark while the intact bone will remain white. During a regular X-ray procedure, a stationary machine sends X-rays through the body to make a single shadow picture. A computed tomography (CT) scan uses computers and rotating X-ray machines to make many successive images (called Figure 1.15 Relative arrangement tomograms) of the inside of body along different directions. In CT scan, the X-ray and motion of X-ray source and source and the detectors rotate simultaneously in opposite direction as shown in detectors in CT scan. Figure 1.18. A motorized table moves the patient (Figures 1.15 and 1.16) through a circular opening in the CT imaging system. As the patient passes through the CT imaging system, a source of X-rays rotates around the inside of the circular opening while the detectors on the other side of the patient record the X-rays exiting the section of the patient’s body being irradiated. The CT scan images provide more detailed information than normal X-ray. Discussion question 1.6 What is the basic difference between conventional X-ray and CT scan? Figure 1.16 Image of CT scan machine 1.3 Medical physics 15 Clinical uses of sound: stethoscope and ultrasound The most familiar clinical use of sound is in the analysis of body sounds with a Brainstorming stethoscope (Figure 1.17). This instrument consists of a small bell-shaped cavity question 1.11 attached to a hollow flexible tube. The bell is placed on the skin over the source Have you ever of the body sound (such as the heart, intestines, or lungs). The sound is then brought your conducted by the pipe to the ears of the examiner who evaluates the functioning ears very near to of the organ. The stethoscope was developed in 1816 by a French physician, Rene a person’s chest? Laennec. A stethoscope can be used to listen sounds made by the heart, lungs or What sounds could you hear? What intestines, as well as blood flow in arteries and veins. A stethoscope can detect simple equipment sound waves with frequency ranging from tens to thousands of Hertz. do medical doctors use to listen these Ultrasound sounds? What information could If the frequency of sound is higher than 20 KHz (0.02 MHz), it is called ultrasonic doctors get from or ultrasound. Typical frequencies used in medical ultrasound are 3.5-10 MHz. these sounds? Ultrasonic waves penetrate tissue and are reflected, scattered and absorbed within it. The scattered and reflected ultrasound contains information about the form and structure of the tissue. Figure 1.17 Stethoscopes Figure 1.18 Ultrasound image showing hyperechoic, hypoechoic and anechoic regions. An ultrasound machine sends an ultrasound wave into a body tissue and detects the reflected wave. The detector generates a tiny electric current that is amplified 16 Unit 1 Application of physics in other fields to generate an ultrasound image on the monitor. An ultrasound image is commonly described by three words: anechoic, hypoechoic and hyperechoic as shown in Fig. 1.18. Anechoic These areas appear black on ultrasound because they do not send back any sound waves (echoless region). Anechoic regions are resulted from fluid-filled regions. Hypoechoic Gives off fewer echoes; These areas appear dark gray because they don’t send back a lot of sound waves (echoes). Hyperechoic These areas bounce back many sound waves. They appear as light gray on the ultrasound image. Brainstorming Radiation therapy question 1.12 The photons of X-rays and gamma-rays and the particles emitted by radioactive Have you ever nuclei all have energies far greater than the energies that bind electrons to atoms heard a medical and molecules. As a result, when such radiation penetrates into biological treatment curing internal body cancer materials, it can rip off electrons from the biological molecules and produce without surgery and substantial alterations in their structure. medicine? How can be possible to do In controlled doses radiation can be used therapeutically. In the treatment of this? certain types of cancer, an ampul containing radioactive material such as radium or cobalt 60 is implanted near the cancerous growth. By careful placement of the radioactive material and by controlling the dose, the cancer cell can be destroyed without greatly damaging the healthy tissue. An externally applied beam of gamma rays or X-rays can also be used to destroy cancerous tumors. The advantage here is that the treatment is administered without surgery 1.19. The effect of radiation on the healthy tissue can be reduced by frequently altering the direction of the beam passing through the body. The tumor is always in the path of the beam, but the dosage received by a given Figure 1.19 Radiotherapy of kidney cancer. section of healthy tissue is reduced. 1.4 Physics and defense technology 17 1.4 Physics and defense technology Brainstorming At the end of this section, you will be able to: question 1.13 List different defense technologies What do physicists do in military? Explain how physics is used in radar, missile and infra-red detection for night vision The modern defense force has different branches like Air Force, Army, Navy and Space Force. All of these defense forces demand different knowledge and advancement of physics like laser guidance and satellite technology, modern electronics, optics, sensing systems, high-energy-density physics, atomic and nuclear physics, hydrodynamics, and physics of advanced materials. The Navy demands oceanographic physics, the propagation of sound through water, deep-ocean currents, and meteorology. Air Force demands turbulent fluid flows, navigation, long-range observation, and pattern recognition. The Army force Brainstorming demands night and all-weather vision and techniques for avoiding detection. question 1.14 Advanced optical physics is important in space-based satellite surveillance Dear students you systems. Advanced optical physics is also important in manned and unmanned are familiar with aircraft, in missiles, and even on rifles. the traffic police and traffic light Radar technology to control the ground vehicles The word RADAR is an acronym derived from the phrase RAdio Detection And traffic. Do you have Ranging. It applies to electronic equipment designed for detecting and tracking any knowledge the presence of objects like ships, vehicles, aircraft, missiles, etc which are at of the air traffic certain distances from the location of the radar. It collects the information related control mechanism? Share with your to the object or target like its range (R) and location by radiating electromagnetic colleagues. signal and examining the echo received from the distant object. Let the time taken for the signal to travel from Radar to target and back to Radar be ‘t’. The two-way distance between the Radar and target will be 2R. The range can be calculated using the speed-distance formula with the speed equal to the speed of light(c). distance ct speed = , distance = speed ∗ time; 2R = c ∗ t and R = time 2 As shown in the Figure 1.21, Radar mainly consists of a transmitter and a receiver. It uses the same antenna for both transmitting and receiving the signals. The Figure 1.20 Radar system 18 Unit 1 Application of physics in other fields transmitter transmits the radar signal in the direction of the target. The target reflects this received signal in various directions. The signal, which is reflected back to the antenna is received by the receiver and displayed on the radar display. Discussion question 1.7 What is the basic principle of radar? Figure 1.21 Basic principles of radar Military applications of Radar Radar is mostly used for military purpose and is one of the most important parts of the air defense system. Its major function is to detect target and guiding the defensive and offensive weapons. Radar can also be utilized in civilian applications particularly in controlling air traffic, observation of weather, navigation of ship, environment, sensing from remote areas, observation of planetary, etc. Missiles A missile is a rocket-propelled or jet-propelled weapon designed to deliver an explosive weapon with great accuracy at high speed. Jets get the oxygen to burn fuel from the air while rockets carry their own oxygen. Missiles are different types. The well-known ones are cruise missile and ballistic missile. Cruise missiles are jet-propelled throughout their flights. Ballistic missiles are rocket-powered only in the initial phase of flight, after which they move under the influence of gravity and air resistance following an arc trajectory to the target. It is governed by Newtonian mechanics. The motion of cruise missile can be controlled by altering the thrust (accelerating a mass of gas) from its engine (or engines) to conserve momentum. A missile is a combination of many electronic, digital and mechanical subsystems that perform many operations to guide the missile from its launcher to its target. There is continuous radio communication between the 1.4 Physics and defense technology 19 internal missile controlling unit and the launch controller to track the target and the proper functioning of each unit of the missile. Infra-red wave detection for night vision Human eyes are sensitive to visible light: red, orange, yellow, green, blue, and Brainstorming violet light. Infrared, is just out of range of what the human eye can detect. It is question 1.15 detected by infrared detecting devices. All people, places, and things give off Do you know any infrared light in an amount proportional to their temperature. Infrared (IR) mechanisms that devices will typically use heat emissions to identify objects that cannot be enable us to see detected using available light sources. Infrared vision is used extensively by the objects at dark night military for various purposes like night vision, navigation, hunting, without available hidden-object detection and targeting. Infrared imaging systems like infrared light source? imaging goggles create an electronic image based on the temperature differences in the radiating object; hotter objects appear brighter than cooler objects. You cannot see the actual color of the objects but temperature difference in the target is represented by different colors that are not related to the actual color of the target. Night vision image is green this is because green is the best wavelength for enhancing the natural night vision in humans to see the targets. Discussion question 1.8 What are the major physics concepts used in Radar and infra-red night vision? Activity 1.4 Visit a nearby military establishment and develop a report on instruments and methods applying physics. Figure 1.22 A helicopter as observed by Night Vision Goggles 20 Unit 1 Application of physics in other fields Brain storming 1.5 Physics in communication question 1.16 At the end of this section, you will be able to: List all the communication Explain the working principle of different communication technologies. technologies you Explain the relation of physics and communication technology have ever used. Do you know other This day, our lives would be very difficult without the use of the communication communication technologies like telephone, cell-phone, mobile and computers. Communication technologies you have not yet used? is transferring of information (message) from one point to another. To transfer the information to the receiver, medium of transmission is required. Depending on the communication medium, the communication system is classified as wired and wireless communication system. Wireless communication systems use radio waves, microwaves and infrared waves. Satellite communication and ground wave communication are common examples of wireless communications. The wire communication system uses wire and optical fiber. Discussion All forms of communication technologies demand the knowledge of physics. The question 1.9 demanded physics knowledge depends on the type of message and the medium How wireless of transmission. The knowledge of electromagnetic theory is crucial to communication understand radio waves, microwaves, infrared waves and visible light which are is possible? used in wireless and fiber optics communication. Electricity and magnetism, electrical circuit, energy, electronics and wave phenomena like reflection, diffraction, refraction, interference, rarefaction and compression of wave propagation are also very important. Unit summary Different scientific fields are dependent on and reinforces the other fields. Physics has a strong influence on many scientific developments and many modern sciences arose from physics. The study of matter and electricity in physics is fundamental towards the understanding of concepts in chemistry, such as atomic structure, molecular structure, X-ray diffractions, radioactivity, periodic properties 1.5 Physics in communication 21 of elements, nature of valency, chemical bonds in molecules, crystal structure of solids and others. In biology, the motion of animals and their body parts, flow of blood through blood vessels,sound production and transmission as well as receiving by special living tissues, communication networks in the body, reaction of living tissue with light, the development of scientific instrument to study living cell demands different concepts of physics Astronomers use the physics of light, atomic physics and Newtonian mechanics for the study of astronomical objects. Geologists use basic physics concepts like force, optics, atomic structure, electromagnetic radiation, heat and heat flow, electricity and magnetism, stress and strain, sea waves, acoustics and fluids and fluid flow to study common geological processes and the analytical techniques. Physicists discover facts and laws, develop methods of measurement, determine various constants, propose and work out in detail mathematical theories and hypotheses, etc., while engineers later apply some of these valuable facts and theories to design and build machines, construction, and operation of various practical devices. In Civil Engineering, the laws of physics can tell you about forces, tension, harmonic vibrations and oscillations, tensile strength, elasticity, and all kinds of other concepts that you can use to make calculations about your designing and construction work. As many of the current technologies wouldn’t be existed without physics, many modern physics experiments also wouldn’t be possible without technology. Medical physics is a branch of physics that concerns the applications of principles of physics to medical diagnosing and treating abnormal tissues. The modern medical equipment like X-ray, MRI, CT scan, ultrasound and others are developed by the application of physics knowledge. Defense technologies like Radar and Infra-red night vision uses the physics of electromagnetic waves. 22 Unit 1 Application of physics in other fields Mechanical Engineering need the concepts of physics like mechanics, dynamics, thermodynamics, materials science, structural analysis, and electricity to design aircraft, watercraft, engines, robotics, weapons, cars, pneumatics, hydraulics and others. Electrical and electronic engineers demand the basic knowledge of physics in electromagnetism and semiconductor physics. End of unit questions 1. Explain the following biological process in terms of some physics knowledge. How our brain receives information from the whole body and send information to other body? Which part of our body tells us the temperature of our environment? What instrument is used to measure the exact value of our body temperature? What is the physics in this instrument? What physics knowledge is needed to understand how a sound is created and transmitted to the listener? What energy transformation occurs in human body? What energy transformation occurs in photosynthesis? 2. List as many concepts of physics that can be used in designing modern vehicles. 3. How physics of light is important to study the astronomical objects? 4. As physics is the basis for the development of many technologies, how technologies contribute for the development of science particularly physics? 5. What major physics knowledge is used in defense radar system to detect the enemy target? 6. List at least three modern medical devices and explain their working principles. 1.5 Physics in communication 23 7. List as much physics knowledge and engineering as possible to build a modern building for residence. 8. Identify the difference between the diagnostic and therapeutic medical device. 9. What are the possible applications of radar system?  Unit 2 Brain storming question 2.1 Two-dimensional motion 1. Consider a ball Introduction shot horizontally from a very high Kinematics is the study of motion without considering its causes. For example, building at a high studying the motion of a football without considering what forces cause or speed. Assume that change its motion. Two-dimensional kinematics are simple extensions of the there is no force of one-dimensional kinematics developed for motion in a straight line in Grade 11. gravity acting on the ball. What would This simple extension will allow us to apply physics to many more situations, and the motion of the it will also yield unexpected insights about nature. ball be like? Explain A ball kicked by a football player, the orbital motion of planets, a bicycle its motion? rounding a curve,the rotation of wheels of a car are a few examples of 2. The ball is two-dimension motion. In fact, most motions in nature follow curved paths projected rather than straight lines. Such types of motion along a curved plane are horizontally from described by two-dimensional kinematics. the top of the same building. This time, At the end of this unit, you will be able to: the force of gravity Understand the basic ideas of two-dimensional motions. is acting on the ball. What will the Describe the motion of objects in horizontal and inclined projectiles; motion of the ball be like? Will gravity Describe uniform rotational motion,rotational dynamics and Kepler’s affect the ball’s laws horizontal motion? Will the ball travel a Describe Newton’s law of Universal gravitation. greater (or shorter) horizontal distance Develop pertinent problem-solving skills. due to the influence of gravity? 24 2.1 Projectile motion 25 2.1 Projectile motion At the end of this section, you will be able to: Explain the motion of the projectile with respect to the horizontal and vertical components of its motion. Derive equations related to projectile motion. Discussion Apply equations to solve problems related to projectile motion question 2.1 A projectile is a thrown, fired, or released object that moves only under the Which motion is different from the influence of gravitational force. The projectile accleration is g = 9.8m/s s. Anyone others? Explain who has observed the motion of a ball kicked by a football player (Figure 2.1b) Why? has observed projectile motion. The ball moves in a curved path and returns to a) A ball thrown the ground. Other examples of projectile motion include a cannonball fired from horizontally into the a cannon, a bullet fired from a gun, the flight of a golf ball and a jet of water air. escaping a hose. b) A bullet fired from a gun. c) A javelin thrown by an athlete. d) A bird flying in the air. Figure 2.1 a) A ball thrown horizontally b) A football kicked in a game Projectile motion of an object is simple to analyze if we make three assumptions: 1. The free-fall acceleration is constant over the range of motion, and it is always directed downward. It is the acceleration due to gravity (g)=9.8m/s 2. 2. The effect of air resistance is negligible. 3. The horizontal velocity is constant because the acceleration of the object does not have vertical component. With these assumptions, we find that the path of a projectile, which we call its trajectory, is a parabola as shown in Figure ??. The horizontal and vertical components of a projectile’s motion are completely independent of each other and can be handled separately, with time t as a 26 Unit 2 Two-dimensional motion common variable for both components. Horizontal Projection In this type of motion the projectile is projected horizontally from a certain height as shown in Figure 2.2. Its initial velocity along the vertical direction is zero and it possesses only horizontal velocity at the beginning. As the time progresses, due to the impact of gravity, it acquires the vertical component of velocity (Figure 2.2). Equations for the horizontal component of motion Figure 2.2 The motion of a ball The projectile has zero acceleration along x direction. Therefore, the initial projected horizontally. velocity v0x remains constant throughout the motion. We use constant acceleration motion equations. The final horizontal velocity, v x after a time t is: vx = v0x (constant) The horizontal distance traveled by the projectile at a time t is given by the equation Discussion question 2.2 ∆x = v 0x t (2.1) Assume that an airplane flying Equations of vertical motion horizontally drops a package to a remote The vertical motion is a constant accelerated motion. We use the kinematic village. equations of motion for constant accelerated motion. The final vertical velocity What kind of motion v y after time t is: is performed by the package? v y = v oy + g t (2.2) Draw the trajectory of the package. Where v0y is the initial vertical velocity. As the package hits the ground at the The initial vertical velocity has no downward component (v0y =0). Therefore village, where is the aircraft? v y = gt From the kinematics equations, the vertical displacement, ∆y has a form: 1 ∆y = v 0y t + g t 2 (2.3) 2 But v0y = 0, therefore 2.1 Projectile motion 27 ∆y = 21 gt2 Remember: When you use equations to answer questions on vertical motion, upwards motion is positive (+) and downwards motion is negative (-). Time of flight Activity 2.1 The time of flight is the time taken by the projectile to hit the ground. Place two tennis We know that: balls at the edge of a tabletop. Sharply ∆y = 21 gt2 snap one ball horizontally off the Then table with one hand while gently tapping s 2∆y t= the second ball off g with your other hand. Measure Range the height (y) of The range is the maximum horizontal distance traveled by the projectile. the table and the Once we find the time of flight t, we can solve for the horizontal displacement horizontal distance between the table’s using: edge and the balls ∆x = v0x t landing location (R). Determine the In projectile motion, the time to cover both the x and y displacement is the same. following from your By substituting the total time flight, we get: measurements: a) The time of flight s 2∆y of both tennis R = v0x g balls. Explain your result. b) The initial Example 2.1 horizontal velocity A rifle is aimed horizontally at a target 30m away as shown in Figure 2.3. The bullet of the balls when hits the target 2 cm below the aiming point. they leave the (a) What is the bullet’s time of flight? edge of the table. (b) What is the initial velocity of the bullet? Assume gravity (g) =10m/s 2. Solution: The givens in this question are: ∆X =30 m, ∆Y =2 cm =0.02 m, g=10m/s 2. (a) The equation for the vertical displacement is: 28 Unit 2 Two-dimensional motion 1 2 ∆y = gt 2 1 -0.02 m = (-10)t2 2 The vertical displacement is in the negative direction, which gives: t=0.06 s Since this is the time of impact with the target, the time of flight of the bullet is also Figure 2.3 A bullet fired the same. horizontally. (b) The equation for x−motion is: ∆x =vox t ∆x 30m vox = = t 0.06s The initial velocity of the bullet is 500 m/s. Example 2.2 A rescue airplane travelling at 360 km/h horizontally dropps a food package from a height of 300 m when it passes over a car driver stranded in the desert. Assumming (g)=10m/s 2. (a) How long will it take the food package to reach the ground? (b) How far from the car driver should the food package be dropped ? Solution: (a) The package has the same horizontal velocity as the airplane. Therefore, the initial vertical velocity is zero. The equation for the vertical displacement is: 1 2 ∆y = gt 2 1 -300 m = (-10)t2 2 The vertical displacement is in the negative direction, which gives: t =7.74 s (b) The equation for the horizontal displacement is: 2.1 Projectile motion 29 ∆x =v0x t ∆x =100 m/s x 7.74 s ∆x =774 m Activity 2.2 Use this activity to investigate horizontal projection. Materials Ruler A cannon ball made from scrunched up aluminum foil. Rubber band. A tube made from paper or cardboard with diameter larger than the diameter of the ball. Procedures: 1. Put the tube near the edge of the table. 2. Use the rubber band to shoot the ball out of the tube. 3. Stretch the rubber band the same amount each time to make sure the initial velocity is constant. 4. You can increase the stretching of the rubber band to increase the initial horizontal velocity of the projectile. 5. Measure the height of the table. Use this height to calculate the time of flight (assume there is no air resistance). 6. Measure the horizontal distance traveled by the canon ball. Use this distance to calculate the initial velocity of the projectile. Inclined projectile motion This is a type of motion in which an object is projected with an initial velocity v0 which makes an angle θ with the horizontal (Figure ??).The initial velocity can be resolved into two components, vertical and horizontal component. The vertical component of the velocity changes with time as a result there is acceleration due 30 Unit 2 Two-dimensional motion to gravity. The horizontal component of the velocity is constant throughout the flight; this is because there is no force acting along the horizontal direction of the projectile as a result there is no acceleration along x-axis. The analysis of the motion involves dealing with the two motions. As shown in Figure 2.4, the projectile has velocity components at different positions. At the top where it reaches its maximum height the vertical Discussion component of the velocity becomes zero. After V y becomes zero the projectile question 2.3 changes its direction and make free fall. Balls A and B are kicked at an angle of 370 and 530 with the horizontal respectively, with the same initial velocity v0. Which ball has: a) the maximum horizontal displacement? b) the maximum height? Figure 2.4 Inclined projectile motion. Equations of inclined projectile motion The initial velocity can be expressed as x component and y component: v0x = v0 cos θ v0y = v0 sin θ The horizontal velocity at any time t is: vx = v0 cos θ (constant) The vertical velocity at any time t is: v y = v o si n θ + g t (2.4) 2.1 Projectile motion 31 Displacements of the projectile There are two different types of displacement of the projectile motion: Horizontal displacement at any time t: ∆x = v 0 cosθt (2.5) Vertical displacement at any time t: 1 ∆y = v 0 si nθt + g t 2 (2.6) 2 The time to reach the maximum height is: v y = v0 sinθ + gt Since v y = 0 at maximum height and g is negative: v 0 sin θ t= g Time of flight The time of flight is the total time for which the projectile remains in flight. The time of flight depends on the initial velocity of the object and the angle of the projection, θ. 1 ∆y = v0 sin θ t + gt 2 2 When the point of projection and point of return are on the same horizontal level, the net vertical displacement of the object is zero, ∆Y =0. 1 0 = v0 sin θ t + gt 2 2 Apply factorization, we have: 0 = t(v0 sin θ + 21 gt) Since t cannot be zero and g is negative,solving for t gives us: 2v 0 sin θ tt ot al = g This last equation does not apply when the projectile lands at a different elevation from the one at which it was launched. 32 Unit 2 Two-dimensional motion Horizontal range and maximum height of a Projectile Let us now consider a special case of projectile motion. Assume a projectile is launched from the origin at O, as shown in Figure 2.4, and returns to the same horizontal level. This situation is common in sports, where baseballs, footballs and golf balls often land at the same level from which they were launched. Two points in this motion are especially interesting to analyze: the peak point A, which has Cartesian coordinates (R/2, H), and the point B, which has coordinates (R, 0). The distance R is called the horizontal range of the projectile, and the distance H is its maximum height. Let us find R and H mathematically in terms of v0 , θ, and g. Range(R) The range of the projectile is the maximum displacement in the horizontal direction. There is no acceleration in this direction since gravity only acts vertically. ∆x = v0 cos θ t When ∆x is maximum, ∆x=R. Since the time to cover the range is the total time of flight: 2v 0 sin θ tt ot al = g R = v0 cos θ tt ot al v 0 2 sin 2θ R= g This equation is valid for launch and impact on a horizontal surface, as shown in Figure 2.5. We can see in Figure 2.5a the range is directly proportional to the square of the initial speed v0 and sin2θ. Furthermore, we can see from the factor sin2θ that the range is maximum at 45◦. In Figure 2.5 (a) we can see that the greater the initial velocity, the greater the range. In Figure (b) the range is maximum at 45◦. This is true only for conditions ignoring air resistance. It is interesting that the same range is found for two initial launch angles that add up to 90◦. The projectile launched with the smaller angle has a lower peak than the higher angle, but they both have the same range. 2.1 Projectile motion 33 Figure 2.5 Trajectories of projectiles on leveled ground. (a) The effect of initial velocity v0 on the range of a projectile with a given initial angle. (b) The effect of initial angle θ on the range of a projectile with a given initial speed. Maximum height (H) The maximum height of a projectile trajectory occurs when the vertical component of velocity, v y equals zero. As the projectile moves upwards it goes against gravity, and therefore the velocity begins to decrease. Eventually the vertical velocity will reach zero, and the projectile is immediately accelerated downward under gravity. Thus, once the projectile reaches its maximum height, it begins to accelerate downward. 1 ∆y = v0 sin θ t + gt 2 2 v 0 sin θ The time to cover the maximum height is: t = g When ∆y is maximum, ∆y =H v 0 2 sin2 θ H= 2g Discussion question 2.4 1. A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection? 2. A ball is kicked into the air from the ground at an angle θ with the horizontal. When the ball reaches its highest point, which statement is true ? (a) Both the velocity and acceleration of the ball are zero. 34 Unit 2 Two-dimensional motion (b) Its velocity is not zero, but its acceleration is zero. (c) Its velocity is perpendicular to its acceleration. (d) Its acceleration depends on the angle at which the ball was thrown. Activity 2.3 3. One ball is thrown horizontally. At the same time, a second ball is Use this activity to dropped from the same height. Ignoring air resistance and assuming the investigate inclined ground is level, which ball hits the ground first? Explain why. projection. You need the materials listed Relation between range and maximum height in Activity 2.2. Procedures: Consider a projectile motion as shown in Figure 2.4. The initial velocity of the (a) Adjust the tube projectile is v0 , H is the maximum height and R is its horizontal range. We know at different that the maximum height of the projectile H is given by the equation: angles from the horizontal. (b) As before keep v 0 2 sin2 θ H= the stretching of 2g the rubber band And horizontal range is given by the equation: constant. (c) Vary the angle of v 0 2 sin 2θ projection. R= g (d) Measure the Divide the maximum height of the projectile by the horizontal range. relationship (In the equation, sin2 θ can be written as sinθsinθ, and sin2θ can be written between the angle as 2sinθcosθ). of projection, H si nθ range and = R 4cosθ maximum height R t anθ reached by the H= 4 projectile. Example 2.3 A football player kicks a ball at angle of 370 with the horizontal. The initial velocity of the ball is 40 m/s. a) Find the maximum height reached by the ball. b) Find the horizontal range of the ball. 2.1 Projectile motion 35 Solution: In this problem the given quantities are: v0 = 40 m/s, θ = 370 and g=10 m/s2 a) The maximum height reached is: V0 2 sin2 θ H= 2g (40m/s)2 sin 370 sin 370 H= 2x10m/s 2 H=28.8 m b) The horizontal range is: V0 2 sin 2θ R= g (40m/s)2 sin 740 R= 10m/s R=153.8 m Example 2.4 A ball is kicked from the ground with an initial speed of 25 m/s at an angle of 530 above the horizontal directly toward a wall, as shown in Figure 2.6. The wall is 24 m from the release point of the ball. (a) How long does the ball take to reach the wall? (b) How far above the ground level does the ball hit the wall? (c) What are the horizontal and vertical components of its velocity as it hits the wall? (d) What is the resultant velocity with it hits the wall? Solution: In this problem the given quantities are: ∆x = 24 m, θ =530 , v0 =25 m/s (a) The horizontal displacement of the ball is given by the equation ∆x = v0 cosθt Solving for the time at which ∆x = 24 m: ∆x 24m t= = v 0 cosθ (25m/s)x0.6 t = 1.6 s Thus, the ball reaches the wall 1.6s after being thrown. 36 Unit 2 Two-dimensional motion (b) We can answer this question if we can find the y coordinate of the ball at the time it hits the wall, namely at t = 1.6 s. We need the y equation of motion. 1 ∆y = v0 sin θ t + gt 2 2 1 ∆y = 25 m/s x0.8 x 1.6 s + x(−10m/s 2 )x(1.6s)2 2 ∆y =19.2 m This tells us that the ball hits the wall at 19.2 m above the ground level. (c) The x and y components of the ball’s velocity at the time of impact (t=1.6 s) vx =v0 cosθ vx = 25 m/s x 0.6 vx =15 m/s v y =v0 sinθ +gt Figure 2.6 A ball thrown toward a v y = 25m/s x 0.8 +(−10m/s 2 )x1.6 s wall. v y =4 m/s (d) The resultant velocity is the vector sum of the x and y components. q v= v x2 + v 2y p v= (15m/s)2 + (4m/s)2 v= 15.5 m/s Activity 2.4: Use this activity to investigate inclined projection. In this activity you use the law of conservation of mechanical energy that you learned in grade 11. Materials required Ruler V-shaped track with a shorter launch track. Small ball (e.g. tennis ball). protractor Procedures: 1. Adjust the shorter end of the track to the edge of the surface of a table.