There are two main types of cells - prokaryotic cells that lack a nucleus and eukaryotic cells that have a nucleus. Kingdoms are divided based on whether cells are prokaryotic like Monera or eukaryotic like Protista, Fungi, Plantae, and Animalia. Unicellular organisms are in the kingdoms of Monera and Protista while multicellular organisms are in Fungi, Plantae, and Animalia.
The document discusses taxonomy and classification systems. It explains that binomial nomenclature is the scientific naming system developed by Carolus Linnaeus, using a two-part Latin name of genus and species. An example is provided of the full taxonomic classification of humans from the broadest kingdom to the specific species. Key terms like genus, species, and DNA are defined. A brief history of classification systems mentions Ernst Haeckel's original three kingdom system.
Osmosis is the diffusion of water across a selectively permeable membrane according to concentration gradients. Cells placed in hypotonic solutions will swell as water enters, while hypertonic solutions cause water to leave cells, shrinking them. Transport across the cell membrane is facilitated by carrier proteins that bind specific molecules and change shape to transport them either passively down gradients or actively against gradients using ATP. Cells take material in through endocytosis and release it through exocytosis using membrane-bound vesicles.
The document discusses nitrogen fixation, which is the process by which nitrogen in the atmosphere is converted into a form that can be used by living things. It then provides details about the general characteristics of bacteria, including that they are single-celled organisms without organelles, have a cell wall composed of peptidoglycan, can reproduce through binary fission, and can perform both aerobic and anaerobic respiration. Finally, it lists 13 biological importance of bacteria, such as their role in producing antibiotics, decomposing waste, and processing foods like cheese and yogurt.
1. Mitosis involves cell division for growth, healing, and asexual reproduction where one diploid cell divides into two identical diploid daughter cells.
2. The main stages of mitosis are prophase, metaphase, anaphase and telophase where chromosomes condense and align before separating and distributing into two new daughter cells.
3. Meiosis involves two cell divisions that result in four haploid cells from one original diploid cell. The main stages are similar to mitosis but with additional processes like independent assortment and crossing over during prophase I and a second division to separate sister chromatids.
The plasma membrane separates the interior of the cell from the outside environment. It is a phospholipid bilayer with embedded proteins. The fluid mosaic model describes the plasma membrane structure, with hydrophobic lipid tails facing the interior and hydrophilic heads facing the exterior. Membrane proteins have various functions like transport, receiving signals, and acting as enzymes. The plasma membrane is selectively permeable, allowing small uncharged molecules to pass through by passive or active transport mechanisms.
Monera is a kingdom composed solely of prokaryotic organisms like bacteria and cyanobacteria. Bacteria can exist as single cells, filaments, or colonies, and can be aerobic or anaerobic, autotrophic or heterotrophic. Other groups in the kingdom include cyanobacteria, which can perform photosynthesis, and eubacteria, which rely on organic carbon sources. Archaebacteria are the most ancient group and differ so greatly from other Monera that they may constitute their own kingdom.
The document discusses the structure and function of prokaryotic cells. Prokaryotic cells are much smaller and simpler than eukaryotic cells, lacking organelles like the nucleus and mitochondria. The basic structure of a prokaryotic cell includes an outer cell wall and cytoplasmic membrane, cytoplasm containing ribosomes and other inclusions, and sometimes additional structures like flagella, pili, or plasmids. The cell wall provides shape and rigidity while the membrane is semipermeable and controls movement of molecules in and out. Ribosomes perform protein synthesis and mesosomes are involved in respiration.
Homeostasis refers to the maintenance of a stable internal environment in an organism. Most homeostatic mechanisms operate through negative feedback loops. A negative feedback loop has components that sense a deviation from the optimal range and trigger responses that act to reverse the deviation and restore homeostasis. An example is the regulation of body temperature through sweating or shivering in response to temperature changes.
In multi-celled organisms, growth occurs through the production of new cells, which increases the total number of cells and total surface area to volume ratio. Metamorphosis in animal development allows for different forms adapted to different environments and food sources over an organism's lifespan. This provides advantages like access to new resources and habitats as the organism matures.
Biology is the science of life and living organisms, including their structure, function, growth, origin, and distribution. It includes subdisciplines like microbiology, botany, zoology, anatomy, physiology, cell biology, biochemistry, genetics, and molecular biology. Biology seeks to understand life at different scales, from molecules to cells to tissues to whole organisms, and examines the chemistry and physics underlying living things. Key concepts that guide biologists include different levels of biological organization and the idea that the difference between living and non-living things is one of degree rather than kind.
Fungi have several key characteristics:
1) They have cell walls made of chitin and obtain nutrients as heterotrophs and decomposers.
2) Their bodies are made of long filaments called hyphae that form a mycelium.
3) They reproduce both sexually through the mating of hyphae and asexually through spores.
The document then discusses the five major fungal phyla - Zygomycota, Oomycota, Ascomycota, Basidiomycota, and Deuteromycota. It provides more details on Zygomycota, including how bread mold is an example and goes through its life cycle and
The document discusses the structure and function of eukaryotic cells. It outlines several key organelles including the cell wall, cell membrane, endoplasmic reticulum, ribosomes, mitochondria, which provide shape, protection, transport, protein synthesis, and energy production respectively to the plant cell.
DNA is composed of nucleotides that contain a phosphate group, five-carbon sugar called deoxyribose, and nitrogenous bases including adenine, guanine, thymine and cytosine. The nucleotides are linked by the phosphate group to form the backbone of the DNA double helix structure.
The document discusses the structures and functions of eukaryotic cells. It outlines the major organelles found in plant and animal cells including the cytoplasm, endoplasmic reticulum, ribosomes, mitochondria, Golgi apparatus, lysosomes, chloroplasts, centrioles, vacuoles, plastids, leucoplasts, chromoplasts, nuclei, nucleoli, nuclear membranes, and chromosomes. It provides details on the specific roles of these organelles in photosynthesis, digestion, cell division, food storage, motility, and genetic functions.
This document discusses the levels of organization of life from cells to the biosphere. It begins by outlining the hierarchical organization from cells to tissues, organs, organisms, populations, communities, ecosystems, and ultimately the biosphere. It then discusses the reductionist principle of breaking down complex life into more simple compartments like chemicals, cells, tissues, organs, and systems within an organism. The next section defines cells and describes the two main types - prokaryotic and eukaryotic. The last section describes the key characteristics of living organisms such as being made of cells, reproduction, genetic code, growth, obtaining and using energy, responding to the environment, and evolving over time.
Bacteria have simple structures, lack organelles, and have circular DNA plasmids instead of chromosomes. They have strong cell walls that are resistant to environmental changes. Bacteria can move using flagella, cilia, or be non-motile. Some bacteria move to spread between hosts while non-motile bacteria spread through other means. Gram staining determines the type of carbohydrates in the bacterial cell wall, distinguishing between thick peptidoglycan walls of Gram-positive bacteria and thinner complex walls of Gram-negative bacteria.
Bacteria can obtain energy through three main methods - autotrophs produce their own energy using solar or chemical energy, chemotrophs produce energy from chemical reactions, and heterotrophs obtain energy from organic matter. Bacteria also reproduce through binary fission, conjugation where genetic material is exchanged, or forming highly resistant spores.
All prokaryotes are classified into two kingdoms: Archaea and Bacteria. Bacteria include cyanobacteria, which are photosynthetic bacteria that appear blue-green in color. Prokaryotes are single-celled organisms that lack nuclei and are found almost everywhere on Earth in enormous numbers. They are divided into the kingdoms Eubacteria and Archaebacteria.
Unicellular green algae like Chlamydomonas reproduce through binary fission and flagellated zoospores. Algae are classified based on their photosynthetic pigments. Green algae contain chlorophyll a, chlorophyll b, and carotenoids. Spirogyra is a filamentous green alga with cylindrical cells arranged in helical filaments. It reproduces asexually through fragmentation and sexually through scalariform and lateral conjugation. Volvox is a colonial green alga. Red algae exhibit varied multicellular forms and contain the red pigment phycoerythrin.
DNA contains 4 nucleotides - adenine, thymine, cytosine and guanine. These nucleotides pair up with each other - adenine pairs with thymine and cytosine pairs with guanine. Together the nucleotides, sugar and phosphate molecules form the structure of DNA.
A virus is a non-living particle composed of genetic material and protein that can infect living cells. Viruses are tiny, obligate intracellular parasites that cannot reproduce on their own and lack metabolic processes. They contain either DNA or RNA as their hereditary material. Viruses are studied to develop vaccines and antiviral drugs, and used in molecular biology experiments. Viruses have a protein capsid that surrounds their nucleic acid and some have an additional phospholipid envelope. They utilize host cell machinery to replicate through lytic or lysogenic cycles.
The document discusses a company called 3DP that is considering two options - launching a new 3D printer product or selling the patent license. It provides information on the estimated costs of product development and market potential for the product. It also provides details on a potential offer from another company to purchase the patent license. The document asks two questions: 1) Calculate the expected monetary value of the two options and recommend the decision based on financial considerations. 2) Calculate the exchange rate change needed to change the recommended decision and its probability.
The document provides solutions to 5 questions related to probability and statistics for computer science.
Question 1 involves calculating the number of ways an inspector can choose 3 batteries from a carton, including the probabilities of getting defective batteries. Question 2 deals with the probability of letters reaching their destination within 3 days. Question 3 is about the probabilities of two firms getting a project based on whether they bid or not. Question 4 calculates probabilities related to drawing colored balls from an urn. Question 5 finds the probability of certain outcomes occurring in independent trials.
Mr. and Mrs. Leonard Maurice are a young couple with two kids seeking financial planning assistance. They have short, intermediate, and long term financial goals including buying furniture, taking a vacation, supporting their kids' education, and retirement. The advisor created a cash budget, analyzed purchasing a home vs renting, and buying a car vs leasing. Insurance and investment plans were also created. The advisor recommended investing savings in debt and equity mutual funds to meet retirement and other long term goals.
The document contains solutions to three math homework problems:
1) An integral problem involving trigonometric substitutions and integration by parts. The solution finds the integral I=1/2[x cos(ln(x))+xsin(ln(x))]+C.
2) A differential equation problem solved using an integrating factor. The solution finds the particular solution as Y=(2x+1)e^2x + ex + (x+1).
3) An eigenproblem with solutions for the eigenvectors and eigenvalues. The roots are found to be real and equal to 2.
The document provides a marketing plan for introducing an eco-friendly product by a student for a selected company. It outlines the objectives of analyzing the marketing environment, making accurate marketing decisions, and developing an effective marketing plan. It then provides a detailed 15-page outline for a marketing plan for Portakabin, a UK-based modular building company, to introduce a new eco-friendly product. The outline includes sections on company background, SWOT analysis, competition analysis, financial analysis, marketing strategy, promotion strategy, and conclusions.
The document discusses the history and importance of microbiology. It describes experiments conducted by Redi and Pasteur that helped disprove the theory of spontaneous generation by showing that microbes come from other living organisms, not from non-living matter alone. It also discusses how cell theory established that cells are the basic unit of structure and function in living things, and how Koch's postulates helped link specific microbes to diseases. Finally, it examines methods of microbial control like antibiotics and how they influenced human health.
Homeostasis refers to the maintenance of a stable internal environment in an organism. Most homeostatic mechanisms operate through negative feedback loops. A negative feedback loop has components that sense a deviation from the optimal range and trigger responses that act to reverse the deviation and restore homeostasis. An example is the regulation of body temperature through sweating or shivering in response to temperature changes.
In multi-celled organisms, growth occurs through the production of new cells, which increases the total number of cells and total surface area to volume ratio. Metamorphosis in animal development allows for different forms adapted to different environments and food sources over an organism's lifespan. This provides advantages like access to new resources and habitats as the organism matures.
Biology is the science of life and living organisms, including their structure, function, growth, origin, and distribution. It includes subdisciplines like microbiology, botany, zoology, anatomy, physiology, cell biology, biochemistry, genetics, and molecular biology. Biology seeks to understand life at different scales, from molecules to cells to tissues to whole organisms, and examines the chemistry and physics underlying living things. Key concepts that guide biologists include different levels of biological organization and the idea that the difference between living and non-living things is one of degree rather than kind.
Fungi have several key characteristics:
1) They have cell walls made of chitin and obtain nutrients as heterotrophs and decomposers.
2) Their bodies are made of long filaments called hyphae that form a mycelium.
3) They reproduce both sexually through the mating of hyphae and asexually through spores.
The document then discusses the five major fungal phyla - Zygomycota, Oomycota, Ascomycota, Basidiomycota, and Deuteromycota. It provides more details on Zygomycota, including how bread mold is an example and goes through its life cycle and
The document discusses the structure and function of eukaryotic cells. It outlines several key organelles including the cell wall, cell membrane, endoplasmic reticulum, ribosomes, mitochondria, which provide shape, protection, transport, protein synthesis, and energy production respectively to the plant cell.
DNA is composed of nucleotides that contain a phosphate group, five-carbon sugar called deoxyribose, and nitrogenous bases including adenine, guanine, thymine and cytosine. The nucleotides are linked by the phosphate group to form the backbone of the DNA double helix structure.
The document discusses the structures and functions of eukaryotic cells. It outlines the major organelles found in plant and animal cells including the cytoplasm, endoplasmic reticulum, ribosomes, mitochondria, Golgi apparatus, lysosomes, chloroplasts, centrioles, vacuoles, plastids, leucoplasts, chromoplasts, nuclei, nucleoli, nuclear membranes, and chromosomes. It provides details on the specific roles of these organelles in photosynthesis, digestion, cell division, food storage, motility, and genetic functions.
This document discusses the levels of organization of life from cells to the biosphere. It begins by outlining the hierarchical organization from cells to tissues, organs, organisms, populations, communities, ecosystems, and ultimately the biosphere. It then discusses the reductionist principle of breaking down complex life into more simple compartments like chemicals, cells, tissues, organs, and systems within an organism. The next section defines cells and describes the two main types - prokaryotic and eukaryotic. The last section describes the key characteristics of living organisms such as being made of cells, reproduction, genetic code, growth, obtaining and using energy, responding to the environment, and evolving over time.
Bacteria have simple structures, lack organelles, and have circular DNA plasmids instead of chromosomes. They have strong cell walls that are resistant to environmental changes. Bacteria can move using flagella, cilia, or be non-motile. Some bacteria move to spread between hosts while non-motile bacteria spread through other means. Gram staining determines the type of carbohydrates in the bacterial cell wall, distinguishing between thick peptidoglycan walls of Gram-positive bacteria and thinner complex walls of Gram-negative bacteria.
Bacteria can obtain energy through three main methods - autotrophs produce their own energy using solar or chemical energy, chemotrophs produce energy from chemical reactions, and heterotrophs obtain energy from organic matter. Bacteria also reproduce through binary fission, conjugation where genetic material is exchanged, or forming highly resistant spores.
All prokaryotes are classified into two kingdoms: Archaea and Bacteria. Bacteria include cyanobacteria, which are photosynthetic bacteria that appear blue-green in color. Prokaryotes are single-celled organisms that lack nuclei and are found almost everywhere on Earth in enormous numbers. They are divided into the kingdoms Eubacteria and Archaebacteria.
Unicellular green algae like Chlamydomonas reproduce through binary fission and flagellated zoospores. Algae are classified based on their photosynthetic pigments. Green algae contain chlorophyll a, chlorophyll b, and carotenoids. Spirogyra is a filamentous green alga with cylindrical cells arranged in helical filaments. It reproduces asexually through fragmentation and sexually through scalariform and lateral conjugation. Volvox is a colonial green alga. Red algae exhibit varied multicellular forms and contain the red pigment phycoerythrin.
DNA contains 4 nucleotides - adenine, thymine, cytosine and guanine. These nucleotides pair up with each other - adenine pairs with thymine and cytosine pairs with guanine. Together the nucleotides, sugar and phosphate molecules form the structure of DNA.
A virus is a non-living particle composed of genetic material and protein that can infect living cells. Viruses are tiny, obligate intracellular parasites that cannot reproduce on their own and lack metabolic processes. They contain either DNA or RNA as their hereditary material. Viruses are studied to develop vaccines and antiviral drugs, and used in molecular biology experiments. Viruses have a protein capsid that surrounds their nucleic acid and some have an additional phospholipid envelope. They utilize host cell machinery to replicate through lytic or lysogenic cycles.
The document discusses a company called 3DP that is considering two options - launching a new 3D printer product or selling the patent license. It provides information on the estimated costs of product development and market potential for the product. It also provides details on a potential offer from another company to purchase the patent license. The document asks two questions: 1) Calculate the expected monetary value of the two options and recommend the decision based on financial considerations. 2) Calculate the exchange rate change needed to change the recommended decision and its probability.
The document provides solutions to 5 questions related to probability and statistics for computer science.
Question 1 involves calculating the number of ways an inspector can choose 3 batteries from a carton, including the probabilities of getting defective batteries. Question 2 deals with the probability of letters reaching their destination within 3 days. Question 3 is about the probabilities of two firms getting a project based on whether they bid or not. Question 4 calculates probabilities related to drawing colored balls from an urn. Question 5 finds the probability of certain outcomes occurring in independent trials.
Mr. and Mrs. Leonard Maurice are a young couple with two kids seeking financial planning assistance. They have short, intermediate, and long term financial goals including buying furniture, taking a vacation, supporting their kids' education, and retirement. The advisor created a cash budget, analyzed purchasing a home vs renting, and buying a car vs leasing. Insurance and investment plans were also created. The advisor recommended investing savings in debt and equity mutual funds to meet retirement and other long term goals.
The document contains solutions to three math homework problems:
1) An integral problem involving trigonometric substitutions and integration by parts. The solution finds the integral I=1/2[x cos(ln(x))+xsin(ln(x))]+C.
2) A differential equation problem solved using an integrating factor. The solution finds the particular solution as Y=(2x+1)e^2x + ex + (x+1).
3) An eigenproblem with solutions for the eigenvectors and eigenvalues. The roots are found to be real and equal to 2.
The document provides a marketing plan for introducing an eco-friendly product by a student for a selected company. It outlines the objectives of analyzing the marketing environment, making accurate marketing decisions, and developing an effective marketing plan. It then provides a detailed 15-page outline for a marketing plan for Portakabin, a UK-based modular building company, to introduce a new eco-friendly product. The outline includes sections on company background, SWOT analysis, competition analysis, financial analysis, marketing strategy, promotion strategy, and conclusions.
The document discusses the history and importance of microbiology. It describes experiments conducted by Redi and Pasteur that helped disprove the theory of spontaneous generation by showing that microbes come from other living organisms, not from non-living matter alone. It also discusses how cell theory established that cells are the basic unit of structure and function in living things, and how Koch's postulates helped link specific microbes to diseases. Finally, it examines methods of microbial control like antibiotics and how they influenced human health.
Transcript - Delta Lake Tips, Tricks & Best Practices (1).pdfcarlyakerly1
This session takes you back to the core principles for for successfully utilizing and operating Delta Lake. We break down the fundamentals—Delta Lake’s structure, transaction management, and data retention strategies—while showcasing its powerful features like time travel for seamless rollback and vacuuming for efficient cleanup.
Demonstrations will teach you how to create and manage tables, execute transactions, and optimize performance with proven techniques. Walk away with a clear understanding of how to harness Delta Lake’s full potential for scalable, reliable data management.
Speakers: Scott Haines (Nike) & Youssef Mirini (Databricks)
YouTube video: https://www.youtube.com/live/O8_82Cu6NBw?si=--4iJL1NkzEPCBgd
Slide deck from presentation: https://www.slideshare.net/slideshow/delta-lake-tips-tricks-and-best-practices-wip-pptx/277984087
Navigating common mistakes and critical success factors
Is your team considering or starting a database migration? Learn from the frontline experience gained guiding hundreds of high-stakes migration projects – from startups to Google and Twitter. Join us as Miles Ward and Tim Koopmans have a candid chat about what tends to go wrong and how to steer things right.
We will explore:
- What really pushes teams to the database migration tipping point
- How to scope and manage the complexity of a migration
- Proven migration strategies and antipatterns
- Where complications commonly arise and ways to prevent them
Expect plenty of war stories, along with pragmatic ways to make your own migration as “blissfully boring” as possible.
Are you spending too much time pulling data, fixing AP delays, and manually processing reports in QuickBooks?
You’re not alone. Many finance teams hit a point where QuickBooks holds them back more than it helps.
The good news – there’s a better way.
Those who moved off QuickBooks instantly noticed how fast they can close the books, automate their cumbersome practices, and be able to create custom reports.
Join us for a 30-minute virtual Lunch & Learn where we’ll break down what it really means to outgrow QuickBooks, and how to take the next step with confidence.
During this session, you’ll learn:
The top signs it’s time to graduate from QuickBooks
Common challenges finance teams face and how modern ERPs solve them
Tips to evaluate and select a more comprehensive ERP system
QuickBooks vs. Cloud ERPs: A side-by-side look
Live Q&A to get all your questions answered
Ready to take the first step toward more automation, faster close, and better reporting?
Tired of Tedious Prompting? Writix AI uses 300+ pre-built workflows so you ca...SOFTTECHHUB
Today’s digital work can feel repetitive and frustrating. Many of us struggle with endless prompt cycles that drain energy and time. Writix AI comes as a breath of fresh air. This tool has been designed to eliminate the constant cycle of trial and error. Instead, you simply select a ready-made workflow, fill in the required details, and let the AI deliver outstanding results in nearly a minute.
Bay Area Apache Spark ™ Meetup: Upcoming Apache Spark 4.0.0 Releasecarlyakerly1
Covering new features and enhancements in the upcoming Apache Spark™ 4.0 release. This deck has an overview of the following features:
✅ Spark Connect: The future of Spark extensibility
✅ ANSI Mode: For better ANSI SQL compatibility
✅ Variant data types for semi-structured data
✅ String collation support
✅ Python UDTF functions
✅ SQL and UDTF functions
✅ PySpark UDF Unified Profiler
Buckeye Dreamin 2024: Assessing and Resolving Technical DebtLynda Kane
Slide Deck from Buckeye Dreamin' 2024 presentation Assessing and Resolving Technical Debt. Focused on identifying technical debt in Salesforce and working towards resolving it.
Managing Changing Data with FME: Part 2 – Flexible Approaches to Tracking Cha...Safe Software
Your data is always changing – but are you tracking it efficiently? By using change detection methods in FME, you can streamline your workflows, reduce manual effort, and boost productivity.
In Part 1, we explored a basic method for detecting changes using the ChangeDetector transformer. But what if your use case requires a more tailored approach?
In this webinar, we’ll go beyond basic comparison and explore more flexible, customizable methods for tracking data changes.
Join us as we explore these three methods for tracking data changes:
- Filtering by modification date to instantly pull updated records.
-Using database triggers in shadow tables to capture changes at the column level.
-Storing all changes in a transaction log to maintain a history of all changes with transactional databases.
Whether you’re handling a simple dataset or managing large-scale data updates, learn how FME provides the adaptable solutions to track changes with ease.
Graph Matrices and Application: Motivational overview, matrix of graph, relat...Rajalingam Balakrishnan
Graph Matrices and Application: Motivational overview, matrix of graph, relations, power of a matrix, node reduction algorithm, building tools. (Student should be given an exposure to a tool like JMeter or Win-runner)
A11y Webinar Series - Level Up Your Accessibility Game_ A11y Audit, WCAG, and...Julia Undeutsch
Are you ready to level up your accessibility knowledge? In this session, we’ll walk through my A11y audit template, learn how it’s helped me understand WCAG guidelines, and discover how you can use it to make impactful changes. I'll take a quick detour into how A11y can help you become active in open source, and how open source can help you improve your a11y skills.
Laura Wissiak will also join the session, and together we’ll dive deep into the POUR principles (Perceivable, Operable, Understandable, Robust) and discuss how to turn audit results into meaningful, actionable tickets that improve accessibility.
With her Pokédex of accessibility you will understand why aiming for AAA accessibility standards isn’t just a goal—it’s about striving for the best, just like in video games. Why play to just pass when you can master the game? Let’s elevate our accessibility efforts together!
Focus: A11y Audit, WCAG 2.1, POUR, Ticketing, Open Source
Target audience: Everyone (Project Managers, Designers, Developers, Testers, and Pokémon Lovers)
A11y Webinar Series - Level Up Your Accessibility Game_ A11y Audit, WCAG, and...Julia Undeutsch
Physics homework help
1. Physics Problem | Sample Assignment | www.expertsmind.com
Question: A game that friends and I try to play to guess how long it takes a ball to fall to the
bottom of a cone and how many revolutions around the cone it will take. I do not have a lot
of physics experience, but I was under the impression that if we knew certain constants such
as: the diameter and height of the cone, the mass of the ball, the velocity of the ball as it
falls into the cone, the angle of the side of the cone, and the usual constants for gravity and
surface friction - that we should be able to know the exact time the ball will fall to the
bottom. We should also be able to know the distance the ball travelled going around the
cone before hitting the bottom. Sadly, I am not quite able to work out how to guess either
the time or distance. What are the formulas to calculate such things? Any suggestions?
Since ball is falling from circumference of Cone at poison M (RED BALL) with initial velocity
v’. It is given that h is the height of the Cone and q is the angle between surface and slant
height of the cone. Its look like as
2. Thus
Height
sin q =
Hypotensis
h
=> sin q =
slant height
=> slant height = h cos ec q
At any point in the path
3. Thus we throw a ball from point M with initial velocity v. then
v ' = vx i + v 'y j
'
Where v x and v 'y are velocities along x and y- direction both.
'
Thus
v ' = v 'cos(f - q )i + v 'sin(f - q ) j
Since motion starts at M point which is h height from the base, therefore position of the Ball
after t time
x = v 'cos(f - q )t
1 2
y - h = v 'sin(f - q )t - gt
2
Eliminating t from these two equations
g
y - h = x tan(f - q ) - x2
2v ' cos (f - q )
2 2
This is the path of the ball.
Now we will estimate the time travel by ball from M to P via N.
2v 'y 2v 'sin(f - q )
Thus total time travel T = Time travel by ball from M to P = =
g g
4. And using this time, maximum travel by ball
2v 'cos(f - q ).v 'sin(f - q )
R = v 'cos(f - q )T =
g
2v '2 cos(f - q ).sin(f - q ) v '2 sin 2(f - q )
= =
g g