Mr. Lowrie's Science Site
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  • Environmental Systems
    • First Week Stuff
    • Key Terms Glossary
    • Unit 1: Introduction to Environmental Science >
      • Chapter 1: Science and the Environment >
        • Section 1: Understanding Our Environment
        • Section 2: The Environment and Society
      • Chapter 2: Tools of Environmental Science >
        • Section 1: Scientific Methods
        • Section 2: Statistics and Models
        • Section 3: Making Informed Decisions
      • Chapter 3: The Dynamic Earth >
        • Section 1: The Geosphere
        • Section 2: The Atmosphere
        • Section 3: The Hydrosphere and Biosphere
    • Unit 2: Ecology >
      • Chapter 4: The Organization of Life >
        • Section 1: Ecosystems: Everything is Connected
        • Section 2: Evolution
        • Section 3: The Diversity of Living Things
      • Chapter 5: How Ecosystems Work >
        • Section 1: Energy Flow in Ecosystems
        • Section 2: The Cycling of Materials
        • Section 3: How Ecosystems Change
      • Chapter 6: Biomes >
        • Section 1: What is a Biome?
        • Section 2: Forest Biomes
        • Section 3: Grassland, Desert, and Tundra Biomes
      • Chapter 7: Aquatic Ecosystems >
        • Section 1: Freshwater Ecosystems
        • Section 2: Marine Ecosystems
    • Unit 3: Populations >
      • Chapter 8: Understanding Populations >
        • Section 1: How Populations Change in Size
        • Section 2: How Species Interact with Each Other
      • Chapter 9: The Human Population >
        • Section 1: Studying Human Populations
        • Section 2: Changing Population Trends
      • Chapter 10: Biodiversity >
        • Section 1: What is Biodiversity?
        • Section 2: Biodiversity at Risk
        • Section 3: The Future of Biodiversity
    • Unit 4: Water, Air, and Land >
      • Chapter 11: Water >
        • Section 1: Water Resources
        • Section 2: Water Use and Management
        • Section 3: Water Pollution
      • Chapter 12: Air >
        • Section 1: What Causes Air Pollution?
        • Section 2: Air, Noise, and Light Pollution
        • Section 3: Acid Precipitation
      • Chapter 13: Atmosphere and Climate Change >
        • Section 1: Climate and Climate Change
        • Section 2: The Ozone Shield
        • Section 3: Global Warming
      • Chapter 14: Land >
        • Section 1: How We Use Land
        • Section 2: Urban Land Use
        • Section 3: Land Management and Conservation
      • Chapter 15: Food and Agriculture >
        • Section 1: Feeding the World
        • Section 2: Crops and Soil
        • Section 3: Animals and Agriculture
    • Unit 5: Mineral and Energy Resources >
      • Chapter 16: Mining and Mineral Resources >
        • Section 1: Minerals and Mineral Resources
        • Section 2: Mineral Exploration and Mining
        • Section 3: Mining Regulations and Mine Reclamation
      • Chapter 17: Nonrenewable Energy >
        • Section 1: Energy Resources and Fossil Fuels
        • Section 2: Nuclear Energy
      • Chapter 18: Renewable Energy >
        • Section 1: Renewable Energy Today
        • Section 2: Alternative Energy and Conservation
      • Chapter 19: Waste >
        • Section 1: Solid Waste
        • Section 2: Reducing Solid Waste
        • Section 3: Hazardous Wastes
    • Unit 6: Our Health and Future >
      • Chapter 20: The Environment and Human Health >
        • Section 1: Pollution and Human Health
        • Section 2: Biological Hazards
      • Chapter 21: Economics, Policy, and the Future >
        • Section 1: Economics and International Cooperation
        • Section 2: Environmental Policies in the United States
        • Section 3: The Importance of the Individual
  • AP Environmental Science
    • First Week Stuff
    • Unit I: Humans and Sustainability: An Overview >
      • Chapter 1: Environmental Problems, Their Causes, and Sustainability
    • Unit II: Science, Ecological Principles, and Sustainability >
      • Chapter 2: Science, Matter, Energy, and Systems
      • Chapter 3: Ecosystems: What are They and How Do They Work?
      • Chapter 4: Biodiversity and Evolution
      • Chapter 5: Biodiversity, Species Interactions, and Population Control
      • Chapter 6: The Human Population and Its Impact
      • Chapter 7: Climate and Biodiversity
      • Chapter 8: Aquatic Biodiversity
    • Unit III: Sustaining Biodiversity >
      • Chapter 9: Sustaining Biodiversity: Saving Species and Ecosystem Services
      • Chapter 10: Sustaining Terrestrial Biodiversity: Saving Ecosystems and Ecosystem Services
      • Chapter 11: Sustaining Aquatic Biodiversity and Ecosystem Services
    • Unit IV: Sustaining Natural Resources >
      • Chapter 12: Food Production and the Environment
      • Chapter 13: Water Resources
      • Chapter 14: Nonrenewable Mineral Resources
      • Chapter 15: Nonrenewable Energy
      • Chapter 16: Energy Efficiency and Renewable Energy
    • Unit V: Sustaining Environmental Quality >
      • Chapter 17: Environmental Hazards and Human Health
      • Chapter 18: Air Pollution
      • Chapter 19: Climate Disruption
      • Chapter 20: Water Pollution
      • Chapter 21: Solid and Hazardous Waste
      • Chapter 22: Urbanization and Sustainability
    • Unit VI: Sustaining Human Societies >
      • Chapter 23: Economics, Environment, and Sustainability
      • Chapter 24: Politics, Environment, and Sustainability
      • Chapter 25: Environmental Worldviews, Ethics, and Sustainability
  • Chemistry
    • First Week Stuff
    • Matter

Motion in Two and Three Dimensions


Position Vector:  The location of a particle relative to the origin of a coordinate system is given by a position vector r, which in unit-vector notation is:
Picture
Here x i-hat, y j-hat, and z k-hat are the vector components of position vector r, and x, y, and z are its scalar components (as well as the coordinates of the particle).  A position vector is described either by a magnitude and one or two angles for orientation, or by its vector or scalar components.

Displacement:  If a particle moves so that its position vector changes from original r to final r, the particle's displacement, delta r is:
Picture
The displacement can also be written as:
Picture

Average Velocity and Instantaneous Velocity:  If a particle undergoes a displacement delta r in time interval delta t, its average velocity v(sub)avg for that time interval is:
Picture
As you shrink time to zero, average velocity reaches a limit called either velocity or instantaneous velocity v:
Picture
which can be written in unit-vector notation as:
Picture
where v(sub)x = dx/dt, v(sub)y = dy/dt, and v(sub)z = dz/dt.  The instantaneous velocity v of a particle is always directed along the tangent to the particle's path at the particle's position.

Average Acceleration and Instantaneous Acceleration:  If a particle's velocity changes from original velocity to final velocity in time interval delta t, its average acceleration  during the time interval is:
Picture
As the time interval is shrunk to zero, average acceleration reaches a limiting value called either acceleration or instantaneous acceleration a:
Picture
In unit-vector notation,
Picture
where a(sub)x = dv(sub)x/dt, 
​
a(sub)y = dv(sub)y/dt, and a(sub)z = dv(sub)z/dt.
Projectile motion:  Projectile motion is the motion of a particle that is launched with an initial velocity v(sub)0.  During its flight, the particle's horizontal acceleration is zero and its vertical acceleration is free-fall acceleration, -g.  (Upward is taken as a positive direction).  If v(sub)0 is expressed as a magnitude (the speed) and an angle, theta (measured from the horizontal), the particle's equations of motion along the horizontal x axis and vertical y axis are:
Picture
The trajectory (path) or a particle in projectile motion is parabolic and is given by:
Picture
if x(sub)0 and y(sub)0 of the above top two equations are zero.  The particle's horizontal range R, which is the horizontal distance from the launch point to the point at which the particle returns to the launch height, is:
Picture

Uniform Circular Motion:  If a particle travels along a circle or circular arc of radius r at constant speed v, it is said to be in uniform circular motion and has an acceleration a of constant magnitude:
Picture
The direction of a is toward the center of the circle or circular arc, and a is said to be centripetal.  The time for the particle to complete a circle is:
Picture
T is called the period of revolution, or simply the period, of the motion.

Relative motion:  When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B.  The two measured velocities are related by:
Picture
where v(sub)BA is the velocity of B with respect to A.  Both observers measure the same acceleration for the particle:
Picture

Files:
APCM Chapter 4 Presentation
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File Type: ppt
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