Finding the components of a vector independent practice


  • Vectors in Two Dimensions
  • 4.1 Displacement and Velocity Vectors
  • Motion along the x direction has no part of its motion along the y and z directions, and similarly for the other two coordinate axes. Thus, the motion of an object in two or three dimensions can be divided into separate, independent motions along the perpendicular axes of the coordinate system in which the motion takes place. To illustrate this concept with respect to displacement, consider a woman walking from point A to point B in a city with square blocks. The woman taking the path from A to B may walk east for so many blocks and then north two perpendicular directions for another set of blocks to arrive at B.

    How far she walks east is affected only by her motion eastward. Similarly, how far she walks north is affected only by her motion northward. Independence of Motion In the kinematic description of motion, we are able to treat the horizontal and vertical components of motion separately. In many cases, motion in the horizontal direction does not affect motion in the vertical direction, and vice versa. An example illustrating the independence of vertical and horizontal motions is given by two baseballs.

    One baseball is dropped from rest. At the same instant, another is thrown horizontally from the same height and it follows a curved path. A stroboscope captures the positions of the balls at fixed time intervals as they fall Figure.

    Figure 4. Each subsequent position is an equal time interval. Arrows represent the horizontal and vertical velocities at each position. The ball on the right has an initial horizontal velocity whereas the ball on the left has no horizontal velocity. Despite the difference in horizontal velocities, the vertical velocities and positions are identical for both balls, which shows the vertical and horizontal motions are independent.

    It is remarkable that for each flash of the strobe, the vertical positions of the two balls are the same. This similarity implies vertical motion is independent of whether the ball is moving horizontally. Assuming no air resistance, the vertical motion of a falling object is influenced by gravity only, not by any horizontal forces. Careful examination of the ball thrown horizontally shows it travels the same horizontal distance between flashes. This is because there are no additional forces on the ball in the horizontal direction after it is thrown.

    This result means horizontal velocity is constant and is affected neither by vertical motion nor by gravity which is vertical. Note this case is true for ideal conditions only. In the real world, air resistance affects the speed of the balls in both directions. The two-dimensional curved path of the horizontally thrown ball is composed of two independent one-dimensional motions horizontal and vertical. The key to analyzing such motion, called projectile motion, is to resolve it into motions along perpendicular directions.

    Resolving two-dimensional motion into perpendicular components is possible because the components are independent. Summary The position function gives the position as a function of time of a particle moving in two or three dimensions. Graphically, it is a vector from the origin of a chosen coordinate system to the point where the particle is located at a specific time. The displacement vector gives the shortest distance between any two points on the trajectory of a particle in two or three dimensions.

    Instantaneous velocity gives the speed and direction of a particle at a specific time on its trajectory in two or three dimensions, and is a vector in two and three dimensions.

    The velocity vector is tangent to the trajectory of the particle. Displacement can be written as a vector sum of the one-dimensional displacements along the x, y, and z directions. Velocity can be written as a vector sum of the one-dimensional velocities along the x, y, and z directions. Motion in any given direction is independent of motion in a perpendicular direction. Conceptual Questions What form does the trajectory of a particle have if the distance from any point A to point B is equal to the magnitude of the displacement from A to B?

    If the instantaneous velocity is zero, what can be said about the slope of the position function? What is the position vector of the particle?

    Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects. Figure 1. Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. Vectors in Two Dimensions A vector is a quantity that has magnitude and direction.

    Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign.

    In two dimensions 2-d , however, we specify the direction of a vector relative to some reference frame i. Figure 2 shows such a graphical representation of a vector, using as an example the total displacement for the person walking in a city considered in Chapter 3.

    A person walks 9 blocks east and 5 blocks north. The displacement is Figure 3. To describe the resultant vector for the person walking in a city considered in Figure 2 graphically, draw an arrow to represent the total displacement vector D.

    In this example, the magnitude D of the vector is Vector Addition: Head-to-Tail Method The head-to-tail method is a graphical way to add vectors, described in Figure 4 below and in the steps following. The tail of the vector is the starting point of the vector, and the head or tip of a vector is the final, pointed end of the arrow. Figure 4. Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in Figure 2.

    The tail of this vector should originate from the head of the first, east-pointing vector. Step 1. Draw an arrow to represent the first vector 9 blocks to the east using a ruler and protractor. Figure 5. Step 2. Now draw an arrow to represent the second vector 5 blocks to the north. Place the tail of the second vector at the head of the first vector. Figure 6.

    Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail. Step 4. Draw an arrow from the tail of the first vector to the head of the last vector.

    This is the resultant, or the sum, of the other vectors. Figure 7. Step 5. To get the magnitude of the resultant, measure its length with a ruler. Note that in most calculations, we will use the Pythagorean theorem to determine this length. Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. Note that in most calculations, we will use trigonometric relationships to determine this angle.

    The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors. Example 1: Adding Vectors Graphically Using the Head-to-Tail Method: A Women Takes a Walk Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths displacements on a flat field. First, she walks Then, she walks Finally, she turns and walks Figure 8.

    Figure 9. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector. Figure Discussion The head-to-tail graphical method of vector addition works for any number of vectors.

    It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in Figure 12 and we will still get the same solution.

    Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative.

    Vectors can be added in any order. Vector Subtraction Vector subtraction is a straightforward extension of vector addition. Essentially, we just flip the vector so it points in the opposite direction. The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So B is the negative of -B; it has the same length but opposite direction. The order of subtraction does not affect the results.

    Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates. The instructions read to first sail If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up?

    Compare this location with the location of the dock. We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip. Discussion Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition. This is an example of multiplying a vector by a positive scalar.

    Notice that the magnitude changes, but the direction stays the same. For example, if you multiply by —2, the magnitude doubles but the direction changes. Note that division is the inverse of multiplication. The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1. Resolving a Vector into Components In the examples above, we have been adding vectors to determine the resultant vector.

    In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it.

    In most cases, this involves determining the perpendicular components of a single vector, for example the x— and y-components, or the north-south and east-west components.

    For example, we may know that the total displacement of a person walking in a city is This method is called finding the components or parts of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement.

    It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Chapter 3. Most of these involve finding components along perpendicular axes such as north and east , so that right triangles are involved. The analytical techniques presented in Chapter 3.

    Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can. The resultant vector is then drawn from the tail of the first vector to the head of the final vector. How do they differ? The total distance traveled along Path 1 is 7. What is the final displacement of each camper? What other information would he need to get to Sacramento? Under what circumstances can you end up at your starting point?

    More generally, under what circumstances can two nonzero vectors add to give zero? More generally, can two vectors with different magnitudes ever add to zero? Can three or more? You may assume data taken from graphs is accurate to three digits. The various lines represent paths taken by different people walking in a city.

    All blocks are m on a side.

    A person walks 9 blocks east and 5 blocks north. The displacement is Figure 3. To describe the resultant vector for the person walking in a city considered in Figure 2 graphically, draw an arrow to represent the total displacement vector D. In this example, the magnitude D of the vector is Vector Addition: Head-to-Tail Method The head-to-tail method is a graphical way to add vectors, described in Figure 4 below and in the steps following.

    The tail of the vector is the starting point of the vector, and the head or tip of a vector is the final, pointed end of the arrow. Figure 4. Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in Figure 2. The tail of this vector should originate from the head of the first, east-pointing vector.

    Step 1. Draw an arrow to represent the first vector 9 blocks to the east using a ruler and protractor. Figure 5. Step 2. Now draw an arrow to represent the second vector 5 blocks to the north. Place the tail of the second vector at the head of the first vector. Figure 6. Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail.

    Step 4. Draw an arrow from the tail of the first vector to the head of the last vector. This is the resultant, or the sum, of the other vectors. Figure 7. Step 5. To get the magnitude of the resultant, measure its length with a ruler. Note that in most calculations, we will use the Pythagorean theorem to determine this length.

    Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. Note that in most calculations, we will use trigonometric relationships to determine this angle. The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.

    Vectors in Two Dimensions

    Example 1: Adding Vectors Graphically Using the Head-to-Tail Method: A Women Takes a Walk Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths displacements on a flat field.

    First, she walks Then, she walks Finally, she turns and walks Figure 8. Figure 9. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector. Figure Discussion The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added.

    Therefore, we could add the vectors in any order as illustrated in Figure 12 and we will still get the same solution. Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative. Vectors can be added in any order.

    Vector Subtraction Vector subtraction is a straightforward extension of vector addition. Essentially, we just flip the vector so it points in the opposite direction. The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction.

    4.1 Displacement and Velocity Vectors

    So B is the negative of -B; it has the same length but opposite direction. We very often need to separate a vector into perpendicular components. For example, given a vector like A in Figure 1, we may wish to find which two perpendicular vectors, Ax and Ay, add to produce it. Figure 1. The vector A, with its tail at the origin of an x, y-coordinate system, is shown together with its x- and y-components, Ax and Ay.

    These vectors form a right triangle. The analytical relationships among these vectors are summarized below. Ax and Ay are defined to be the components of A along the x— and y-axes. The relationship does not apply for the magnitudes alone.

    However, it is not true that the sum of the magnitudes of the vectors is also equal. To find Ax and Ay, its x— and y-components, we use the following relationships for a right triangle. Suppose, for example, that A is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods.

    Figure 3. Figure 4. The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components Ax and Ay have been determined. Both processes are crucial to analytical methods of vector addition and subtraction.

    Adding Vectors Using Analytical Methods To see how to add vectors using perpendicular components, consider Figure 5, in which the vectors A and B are added to produce the resultant R. Figure 5. Vectors A and B are two legs of a walk, and R is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of R. If A and B represent two legs of a walk two displacementsthen R is the total displacement. The person taking the walk ends up at the tip of R.

    There are many ways to arrive at the same point. In particular, the person could have walked first in the x-direction and then in the y-direction. Those paths are the x— and y-components of the resultant, Rx and Ry. When you use the analytical method of vector addition, you can determine the components or the magnitude and direction of a vector.

    Step 1. Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes. Figure 6. To add vectors A and B, first determine the horizontal and vertical components of each vector.


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