AGE EDITOR(S): Ivana Gonzalez, Kiara Cerda, Louise Fischel-Bock, Rony Krell

Dimensions

  • Dimensions we need to know are: length, mass, and time
  • Some measurements combine dimensions.
    • Velocity = L/T (Length / Time)
    • Acceleration = L/T^2 (Length / Time^2)
  • In order for equations to be valid, each term in a certain equation must have the exact same dimensions.
    • For example: V= 3A (or velocity equals 3*acceleration) is not valid because velocity does not have dimensions identical to acceleration.
    • 2342321.43504(V/T) =392340932840298(A) is valid because each term has identical dimensions. The coefficients or "numbers" before each term do not make a difference in "dimensional validity".
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Vectors


  • A vector is a type of measurement that illustrates a force's magnitude and direction. It is drawn as an arrow (whose length depends on the magnitude of the force) pointed according to where the force is directed. For example, Vector A has a magnitude of 5 Newtons pointed 45 degrees north of east. It is shown like this
  • A vector can have a negative magnitude depending on which direction is perceived as positive. For example, when dropping a ball from above, its magnitude is perceieved as negative. In general, movement towards the origin is perceieved as negative and movement away from the origin is perceived as positive.
  • A vector may have either one or two component vectors (in a two dimensional situation). Each component vector can be created by "shining a flashight" at the vector from a direction (up, down, left, or right) and drawing a straight line on the opposite side to indicate the vector's displacement. By drawing two component vectors one can form a triangle with a vector and calculate magnitudes and angles using trigonometry.
  • In a situation where you must add or subtract vectors, the components of the resultant vector, "Vector R", usually follow the following equation: R(x) = A(x) +/- B(x) +/- C(x) etc. and R(y)= A(y) +/- B(y) +/- C(y) etc. Note that each component's sign must be taken into account.
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    Addition of multiple vectors

  • Figure 2 (a) shows that A + B = B + A. The sum of the vectors is called the resultant and is the diagonal of a parallelogram with sides A and B. Figure 2 (b) illustrates the construction for adding four vectors. The resultant vector is the vector that results in the one that completes the polygon.

3 .

Vector AdditionThe easiest way to learn how vector addition works is to look at it graphically. There are two equivalent ways to add vectors graphically: the tip-to-tail method and theparallelogram method. Both will get you to the same result, but one or the other is more convenient depending on the circumstances.Tip-to-Tail MethodWe can add any two vectors, A and B, by placing the tail of B so that it meets the tip ofA. The sum, A + B, is the vector from the tail of A to the tip of B.external image tiptotail_2.gifNote that you’ll get the same vector if you place the tip of B against the tail of A. In other words, A + B and B + A are equivalent.Parallelogram MethodTo add A and B using the parallelogram method, place the tail of B so that it meets the tail of A. Take these two vectors to be the first two adjacent sides of a parallelogram, and draw in the remaining two sides. The vector sum, A + B, extends from the tails of Aand B across the diagonal to the opposite corner of the parallelogram. If the vectors are perpendicular and unequal in magnitude, the parallelogram will be a rectangle. If the vectors are perpendicular and equal in magnitude, the parallelogram will be a square.external image parallelogram_2.gifAdding Vector MagnitudesOf course, knowing what the sum of two vectors looks like is often not enough. Sometimes you’ll need to know the magnitude of the resultant vector. This, of course, depends not only on the magnitude of the two vectors you’re adding, but also on the angle between the two vectors.Adding Perpendicular VectorsSuppose vector A has a magnitude of 8, and vector B is perpendicular to A with a magnitude of 6. What is the magnitude of A + B? Since vectors A and B are perpendicular, the triangle formed by A, B, and A + B is a right triangle. We can use the Pythagorean Theorem to calculate the magnitude of A + B, which is external image phy.total22.gifexternal image PerpAdd.gifAdding Parallel VectorsIf the vectors you want to add are in the same direction, they can be added using simple arithmetic. For example, if you get in your car and drive eight miles east, stop for a break, and then drive six miles east, you will be 8 + 6 = 14 miles east of your origin. If you drive eight miles east and then six miles west, you will end up 8 – 6 = 2 miles east of your origin.external image parallel.gifAdding Vectors at Other AnglesWhen A and B are neither perpendicular nor parallel, it is more difficult to calculate the magnitude of A + B because we can no longer use the Pythagorean Theorem. It is possible to calculate this sum using trigonometry. EXAMPLE





Vector A has a magnitude of 9 and points due north, vector B has a magnitude of 3 and points due north, and vector C has a magnitude of 5 and points due west. What is the magnitude of the resultant vector, A + B + C?

First, add the two parallel vectors, A and B. Because they are parallel, this is a simple matter of straightforward addition: 9 + 3 = 12. So the vector A + B has a magnitude of 12 and points due north. Next, add A + B to C. These two vectors are perpendicular, so apply the Pythagorean Theorem:external image 04.sqrt+12+2.gifThe sum of the three vectors has a magnitude of 13.


  • external image 10010.nfg006.jpgThis picture illustrates vector subtraction. To subtract A and B you just add positive A and -B. Therefore A - B= D.




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  • This picture illustrates the tail to head method of adding vectors. Vector C is the sum of Vectors A and B.
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If you really don't understand adding vectors with components:
http://www.youtube.com/watch?v=u5iZbY_-kvA
http://www.youtube.com/watch?v=JM4pgrEZDFI&feature=related
Video defines vectors and their purpose. Video also breaks down addition and subtraction of vectors and how one does that.Disregard after 2:10

Error Calculation


  • % error = 100*|actual value - experimental value|/actual value


Websites




Pictures
Vectors forming a cool shape
Vectors forming a cool shape
Straight vectors forming a circle
Straight vectors forming a circle
vector.jpg
demonstrates head to tail addition and the resultant vector c that is formed
...in the spirit of velocity :)
...in the spirit of velocity :)
Vector_Addition.gif
shows both methods for adding vectors: head to tail first and then the parralelogram method

Video

http://www.5min.com/Video/Base-Jumping-in-Terminal-Velocity-97275667

Besides the very cool aerial shots, I chose this video because the people reach terminal velocity (around 120 mph) while flying off ridiculously huge mountains

Natland Note (9/21/09): Good stuff guys. The video isn't as related, but I like it anyway. is it possible to include some pictures showing vector addition? e.g. if you follow this link: http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Kinematics-in-Two-Dimensions.topicArticleId-10453,articleId-10416.html, there is a picture or two showing vector addition. Or you can type "vector addition" into google images and see what you find.

SOURCES**

Vector addition examples are taken from http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Kinematics-in-Two-Dimensions.topicArticleId-10453,articleId-10416.htmlhttp://www.sparknotes.com/testprep/books/sat2/physics/chapter4section2.rhtml