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## Chapter 3 Kinematics in Two or Three Dimensions; Vectors Homework How do we calculate the motion of this skier in two dimensions Adding & Subtracting Vectors

Escabar, Mimi, Features Editor has reference to this Academic Journal, PHwiki organized this Journal Chapter 3 Kinematics in Two or Three Dimensions; Vectors Homework Tuesday: Read Chapter 4 2014: Do P. 77 1,2,4,5,11,12,13,14 2013: Do p. 77 1, 2, 4, 5, 18 2012: Do p. 77 1, 2, 4, 5, 11, 12, 13, 14, 17 How do we calculate the motion of this skier in two dimensions

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How do we calculate the motion of this skier in two dimensions How do we calculate the tension in these ropes 3-1 Vectors in addition to Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, as long as ce, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature

3-2 Addition of VectorsGraphical Methods For vectors in one dimension, simple addition in addition to subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates. 3-2 Addition of VectorsGraphical Methods If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem. 3-2 Addition of VectorsGraphical Methods Adding the vectors in the opposite order gives the same result:

3-2 Addition of VectorsGraphical Methods Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method. 3-2 Addition of VectorsGraphical Methods The parallelogram method may also be used; here again the vectors must be tail-to-tip. 3-3 Subtraction of Vectors, in addition to Multiplication of a Vector by a Scalar In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction. Then we add the negative vector.

3-3 Subtraction of Vectors, in addition to Multiplication of a Vector by a Scalar A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction. Adding & Subtracting Vectors Vectors can be added or subtracted from each other graphically. Each vector is represented by an arrow with a length that is proportional to the magnitude of the vector. Each vector has a direction associated with it. When two or more vectors are added or subtracted, the answer is called the resultant. A resultant that is equal in magnitude in addition to opposite in direction is also known as an equilibrant. Adding Vectors using the Pythagorean Theorem 3 m 4 m + If the vectors occur such that they are perpendicular to one another, the Pythagorean theorem may be used to determine the resultant. A2 + B2 = C2 (4m)2 + (3m)2 = (5m)2 When adding vectors, place the tail of the second vector at the tip of the first vector.

+ Adding & Subtracting Vectors 3 m 4 m – 7 m = If the vectors occur in a single dimension, just add or subtract them. When adding vectors, place the tail of the second vector at the tip of the first vector. When subtracting vectors, invert the second one

How to Solve: P 1. Add vectors by placing them tip to tail. P or 2. Draw the resultant. P This method is also known as the Parallelogram Method. How to Solve: This method is also known as the Parallelogram Method. 3-4 Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.

3-4 Adding Vectors by Components If the components are perpendicular, they can be found using trigonometric functions. Remember: soh cah toa 3-4 Adding Vectors by Components The components are effectively one-dimensional, so they can be added arithmetically. 3-4 Adding Vectors by Components Adding vectors: Draw a diagram; add the vectors graphically. Choose x in addition to y axes. Resolve each vector into x in addition to y components. Calculate each component using sines in addition to cosines. Add the components in each direction. To find the length in addition to direction of the vector, use: in addition to .

3-4 Adding Vectors by Components Example 3-2: Mail carriers displacement. A rural mail carrier leaves the post office in addition to drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east as long as 47.0 km. What is her displacement from the post office 3-4 Adding Vectors by Components Example 3-3: Three short trips. An airplane trip involves three legs, with two stopovers. The first leg is due east as long as 620 km; the second leg is southeast (45°) as long as 440 km; in addition to the third leg is at 53° south of west, as long as 550 km, as shown. What is the planes total displacement Vector vs. Scalar The resultant will always be less than or equal to the scalar value. 670 m 270 m 770 m 868 m dTotal = 1,710 m d = 868 m

Homework 2014: Read in addition to provide notes on 4.2 Do p. 78 27, 28, 29, 30, 31 2013: Do p. 78 21, 23, 27, 28, 29, 30 2012: Do p. 78 21, 23, 27, 28, 29, 30, 31 d = 23 km dx = d cos dx = (23 km)(cos 30°) dx = 19.9 km dy = d sin dy = (23 km)(sin 30°) dy = 11.5 km Example 2: A bus travels 23 km on a straight road that is 30° North of East. What are the component vectors as long as its displacement x d dx dy y Algebraic Addition In the event that there is more than one vector, the x-components can be added together, as can the y-components to determine the resultant vector. Rx = ax + bx + cx Ry = ay + by + cy R = Rx + Ry R

Now do the following on the map: Start at RCK Go 5 cm North Go 10 cm at 10 degrees N of W Go 20 cm at -80 degrees Go 5 cm at 190 degrees Go 18.5cm at 50 degrees (N of E) Where are you

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