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LNK2LRN™ AP Physics C Website Notes and Plans for the Summer of 2010 July 21 to July31 Topic I: Vectors.
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AP Physics C students: Mathematics is the language of physics. This is what allows us to quantitatively describe the world around us. In Mechanics, the study of motion, we will use two types of quantities to represent concepts like acceleration, mass, and time numerically. These two types are known as scalars and vectors. The ability to work with vectors is essential for an understanding of physics. Remember that a vector is a quantity that has two aspects. It has a size, or magnitude, and a direction. In contrast, there are quantities called scalars that have only size.
Very Important: If you have any questions, send an email to rpersin@fau.edu
Summer 2010 Website notes for Scalars and Vectors:
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I. Vectors. Vectors are used to describe multi-dimensional quantities. Multi-dimensional quantities are those which require more than one number to completely describe them. Vectors, unlike scalars, have two characteristics, magnitude and direction. A vector is indicated by an uppercase letter either in boldface or with an arrow over the top. For example, A or  . Examples of vector quantities are: position in a plane, position in space, velocity, acceleration , and force. |
II. Scalars. Scalars are used to describe one- dimensional quantities, that is, quantities which require only one number to completely describe them. They have magnitude only. Direction does not apply. There are cases where scalars can be combined mathematically, but we will save that for later. Some examples of scalar quantities are: temperature, mass, time, volume, density, length, area, and energy. |
III. Addition of Vectors Vectors can be added graphically using the head-to-tail method. You begin by drawing the first vector in a coordinate system, and then drawing the second vector from the endpoint of the first, and so on. Then you draw a single vector from the origin to the head of the last vector.
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IV. Vector Diagram. Any vector can be resolved into perpendicular component vectors using sine and cosine functions. Actually, for all vector problems just remember SOHCAHTOA.
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V. Vector Equations. The Pythagorean theorem and the inverse tangent function can be used to find the magnitude and direction of a resultant vector. The length (magnitude) and components for the vector A, shown in the previous panel, and the angle Ø ("Phi") are computed as:
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With these equations you can
easily show that vectors can be added by summing their x and y components.
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VI. Some Properties of Vectors. Two vectors are equal only if they have the same magnitude and direction. To find the opposite of a given vector just keep the same magnitude but point it in the opposite direction. ex. A - B = A + (-B) Vectors can also be expressed using polar coordinates (r , θ) specifying the length of the radius vector r , and the angle of rotation, Ø ("Phi"), from the positive x-axis.
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VII. Using the Unit Vectors i, j, and k.Additionally, in a two-dimensional coordinate system, vectors can be denoted using the unit vectors î and ĵ. Each unit vector has magnitude 1, and they point in the x and y directions, respectively. We can easily add the third dimension, or z direction using unit vector k.
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VIII. Vector Subtraction.The vector difference works the same as vector addition except that we multiply the vector we are subtracting by -1. It is much like subtracting two numbers: A - B = A + (-B). The diagram below illustrates vector subtraction in the tip-to-tail style. The original B vector is shown as a dotted line.
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Problem Set for Week #1: June 12 to June 20. Send your work and solutions as an email attachment to rpersin@fau.edu . 1. If vector C is added to vector B, the result is –9i – 8j. If B is subtracted from C, the result is 5i + 4 j. What is the direction of B (to the nearest degree)? a. 225° b. 221° c. 230° d. 236° e. 206° 2. Starting from one oasis, a camel walks 25 km in direction 30° south of west and then walks 30 km toward the north to a second oasis. What is the direction from the first oasis to the second oasis? a. 21° N of W b. 39° W of N c. 69° N of W d. 51° W of N e. 42° W of N 3. A hunter wishes to cross a river that is 1.5 km wide and flows with a velocity of 5.0 km/h parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of 12 km/h with respect to the water. What is the minimum time for crossing?
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And Always Remember... "From Newtonian Mechanics, Through Quantum Theory, Without Knowledge of Physics, Life Would Be Dreary." |
IMPORTANT WEBSITES TO VISIT:
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