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Unit 2-3: Vectors, Velocity, and Acceleration

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Vector or Scalar? Measurements and quantities can have two properties: –Magnitude –Direction A scalar quantity only contains magnitude A vector quantity contain both magnitude and direction.

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Vector or Scalar? Scalar quantities include: –Speed –Mass –Volume –Density –Time

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Vector or Scalar? Vector quantities include: –Weight –Forces –Velocity –Acceleration –Current

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Vector or Scalar? We use pictures to represent vector quantities, specifically arrows. The arrow direction indicates…direction. The length of the arrow is drawn to scale and represents the magnitude of the quantity.

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Vector or Scalar Ex: A car traveling to the right at 5m/s and a car traveling to the left at 10m/s Observe how the 10m/s arrow is twice the length of the 5m/s arrow.

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Adding Vectors Vectors can be added like a scalar quantity can be added, however, the direction determine the final answer: In one-dimensional motion: –Forward or up is labeled as positive –Backward or down is labeled as negative Vectors are always added. Sometimes we are just adding a negative quantity.

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Adding Vectors Consider an airplane flying east (positive) at 100km/h: The airplane encounters a wind blowing with the plane (positive) at 20km/h. Add the vectors ‘tip to tail’ –Connect the tip of the first to the tail of the second –Add together –The resultant vector is the result of adding the two magnitudes. (120km/h)

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Adding Vectors Now consider the same airplane flying east (positive) at 100km/h: The airplane now encounters a wind blowing against the plane (negative) at 20km/h. Add the vectors ‘tip to tail’ –The resultant vector again is the result of adding the two magnitudes together. (80km/h)

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Adding Vectors The result vector has the direction of the largest magnitude vector. (positive or negative) The result vector magnitude is the sum of all the magnitudes (with positive and negative directions) added together.

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Adding Vectors Examples: –A runner on a treadmill is running at 10m/s and the treadmill belt is circling at 10m/s in the opposite direction. Total velocity of runner = 0m/s –A boat is traveling downstream at 16m/s with a current of 3m/s. Total velocity of boat = 19m/s –A boat is traveling upstream at 18m/s against a current of 8m/s. Total velocity of boat = 10m/s

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Distance vs. Displacement Our quantity of distance (from the last sections) was a scalar quantity. We were not looking at the direction of motion. When direction is taken into account, you are looking at displacement. Distance is the total area covered while moving Displacement is the direct line path’s length.

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Distance vs. Displacement Observe: A person starts at 0m, and then moves back 2 meters, then moves forward 3m. What is the person’s distance and displacement? Distance = 5m = 2m + 3m Displacement = 1m = -2m +3m A person starts at 0m, and then moves back 3 meters, then moves forward 2m, then moves forwards another 3m, then moves backwards 6m. What is the person’s distance and displacement? Distance = 14m = 3m + 2m + 3m + 6m Displacement = -4m = -3m + 2m + 3m + -6m

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Distance vs. Displacement Distance = the sum of all movement taken Distance: x = x1 + x2 + x3 + … Displacement = the difference between the final position and the initial position. Displacement x = xf – xi

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Speed & Velocity Speed and velocity are both looking at how quickly something is moving, however velocity is also looking at direction. Speed: Scalar (magnitude only) Velocity: Vector (magnitude and direction) Ex: –Traveling at 60km/h (speed) –Traveling north at 60km/h (velocity)

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Velocity Velocity is determined by dividing the displacement by the time taken. velocity = displacement/time v = (xf-xi)/t Example: –You start at 0m, move forward 3m, and move back 1m. This takes 2.4 seconds. Determine your velocity. V = 0.83m/s

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Velocity When motion is only in one dimension, direction is indicated as being in the positive or negative direction, such as on a number line. A negative velocity is pointing in the negative direction, a positive velocity is pointing in the positive direction.

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Constant Velocity From the definition of velocity, having a constant velocity requires constant magnitude speed and constant direction. Constant direction must be in a straight line. Motion at constant velocity is motion in a straight line at constant speed.

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Changing Velocity Velocity changes whenever speed or direction (or both) changes. Constant speed and constant velocity are not the same thing. –You can be traveling at a constant speed while continually changing your direction.

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Acceleration We can change the state of motion of an object by changing its speed or direction of motion (or both). The rate at which velocity changes is called acceleration. Acceleration is a rate-> change over time. Acceleration = change in velocity time

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Acceleration Specifically: Acceleration = Final velocity – Initial velocity Time a = vf – vi t Examples: A car accelerates from 20.0m/s to 33.2m/s in 7.40s. Determine the acceleration. 1.78m/s 2

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Acceleration The unit of acceleration is the meter per second per second, also said as the meter per second squared. Acceleration can also be negative, also referred to as deceleration, this simply means that the object is slowing down.

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Acceleration But if we don’t have final velocity, initial velocity, acceleration, or time, we can substitute and rearrange to get new formulas to use. Looking at these formulas, you choose which one has what you are looking for and what you are not given.

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Formulas v = xa = vf – vi t t x = vit + ½ at 2 Vf 2 = vi 2 + 2ax

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Examples Determine the acceleration of a car that accelerates from rest to 20.0m/s in 4 seconds. 5m/s 2 A car is traveling at 30.0m/s and slams on the brakes. It takes 4.5 seconds to stop. Determine the acceleration. -6.7m/s 2

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Examples A car accelerates from 8.99m/s at 2.3m/s 2 for 3.00s. How far will the car travel? What is the final velocity of the car? –X = 37m –Vf = 16m/s

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Examples A sprinter is on a 1.00x10 2 m track and accelerates at 1.22m/s 2 to his top speed in 4.56s. What is is starting speed? A car travels 32kilometers. The car started at 8.55km/h and accelerates to 25km/h. Determine the car’s acceleration.

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