Transcript vectorsx
Vectors Vectors and scalars A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars. Vector Has magnitude and direction Ex: displacement, velocity, acceleration, and force Scalar Magnitude only Ex: speed, distance Vector quantities ”A pokemon is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the east of north." This statement now provides a complete description of the displacement vector - it lists both magnitude (20 meters) and direction (30 degrees to the east of north) relative to a reference or starting position (the center of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed. Represent vectors Free body diagrams Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Head to tail method Directions The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from east, west, north, or south. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction). The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "tail" from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit. Magnitude of a vector The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. A scale should be drawn along with a vector to show it’s magnitude. 5 cm= 2 miles Vector addition Two vectors can be added together to determine the result (or resultant) There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are: the Pythagorean theorem and trigonometric methods the head-to-tail method using a scaled vector diagram The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. Theorem continued Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. This problem asks to determine the result of adding two displacement vectors that are at right angles to each other. The result (or resultant) of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant (i.e., the hypotenuse of the right triangle). Using Trigonometry to Determine a Vector's Direction The direction of a resultant vector can often be determined by use of trigonometric functions SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Trigonometry continued These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. The process begins by the selection of one of the two angles (other than the right angle) of the triangle. Once the angle is selected, any of the three functions can be used to find the measure of the angle. Write the function and proceed with the proper algebraic steps to solve for the measure of the angle. Important to check what Quadrant you are in! The Calculated Angle is Not Always the Direction The following vector addition diagram is an example of such a situation. Observe that the angle within the triangle is determined to be 26.6 degrees using SOH CAH TOA. This angle is the southward angle of rotation that the vector R makes with respect to West. Yet the direction of the vector as expressed with the CCW (counterclockwise from East) convention is 206.6 degrees. Head to tail he head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, head-totail method). The process is repeated for all vectors that are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East. Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper. Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m). Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram. Repeat steps 2 and 3 for all vectors that are to be added Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as Resultant or simply R. Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m). Measure the direction of the resultant using the counterclockwise convention discussed earlier in this lesson. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg. SCALE: 1 cm = 5 m Distance and displacement Distance Scalar (How much ground was covered) Displacement Vector (How far out the object has gone) (where it was displaced) CHANGE IN POSITION Test this theory A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. Even though the physics teacher has walked a total distance of 12 meters, her displacement is 0 meters Explanation During the course of her motion, she has "covered 12 meters of ground" (distance = 12 m). Yet when she is finished walking, she is not "out of place" - i.e., there is no displacement for her motion (displacement = 0 m). Displacement, being a vector quantity, must give attention to direction. The 4 meters east cancels the 4 meters west; and the 2 meters south cancels the 2 meters north. Vector quantities such as displacement are direction aware. Scalar quantities such as distance are ignorant of direction. In determining the overall distance traveled by the physics teachers, the various directions of motion can be ignored Difference between distance and displacement a vector quantity such as displacement is direction-aware a scalar quantity such as distance is ignorant of direction. When an object changes its direction of motion, displacement takes this direction change into account Heading the opposite direction effectively begins to cancel whatever displacement there once was. Problem 1. What is the displacement of the cross-country team if they begin at the school, run 10 miles and finish back at the school? Answer The displacement of the runners is 0 miles. While they have covered a distance of 10 miles, they are not "out of place" or displaced. They finish where they started. Round-trip motions always have a displacement of 0. Question 2 2. What is the distance and the displacement of the race car drivers in the Indy 500? Answer 2. The displacement of the cars is somewhere near 0 miles since they virtually finish where they started. Yet the successful cars have covered a distance of 500 miles. Speed and Velocity Speed Scalar (How fast moving) Velocity Vector the rate at which an object changes its position." What would be the velocity? Imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. Answer While this might result in a frenzy of activity, it would result in a zero velocity. Explanation Since velocity is defined as the rate at which the position changes, this motion results in zero velocity. velocity If a person in motion wishes to maximize their velocity, then that person must make every effort to maximize the amount that they are displaced from their original position. Turning around would null the effort Velocity Velocity is a vector quantity= must have ______________ and _______________________ For instance, you must describe an object's velocity as being 55 mi/hr, east. This is one of the essential differences between speed and velocity. Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity and is direction aware. Velocity The direction of the velocity vector is simply the same as the direction that an object is moving. Does not rely on speeding up or slowing down Calculating Average Speed and Average Velocity As an object moves, it often undergoes changes in speed. One minute you are traveling 50 mph the next you are at 70 mph. The average speed during the course of a motion is often computed using the following formula: Average speed= distance traveled/ time of travel Average velocity Average velocity = △ position / time = displacement / time Problem 1 While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8 hours. What was her average speed?