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**Two-Dimensional Motion and Vectors**

Chapter 3 pg

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What do you think? How are measurements such as mass and volume different from measurements such as velocity and acceleration? How can you add two velocities that are in different directions? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Some students will be able to deduce the answer to the first question based on their work with the previous chapter. Some measurements (such as mass and volume) do not include direction, while other measurements (such as velocity and acceleration) do. After discussing this, have students list other types of measurements, and determine whether they each one includes a direction. Students may not be able to answer the second question (unless they have covered this in an earlier science course), but it will help motivate them to learn the upcoming material.

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**Introduction to Vectors**

Scalar - a quantity that has magnitude but no direction Examples: volume, mass, temperature, speed Vector - a quantity that has both magnitude and direction Examples: acceleration, velocity, displacement, force Emphasize that direction means north, south, east, west, up, or down. It does not mean increasing or decreasing. Even though the temperature may be going “up”, it is really increasing and has no direction. To further emphasize the distinction, point out that it is meaningless to talk about the direction of temperature at a particular point in time, while measurements such as velocity have direction at each moment.

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**Vector Properties Vectors are generally drawn as arrows.**

Length represents the magnitude Arrow shows the direction Resultant - the sum of two or more vectors Make sure when adding vectors that You use the same unit Describing similar quantities

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**Finding the Resultant Graphically**

Method Draw each vector in the proper direction. Establish a scale (i.e. 1 cm = 2 m) and draw the vector the appropriate length. Draw the resultant from the tip of the first vector to the tail of the last vector. Measure the resultant. The resultant for the addition of a + b is shown to the left as c. Ask students if a and b have the same magnitude. How can they tell?

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Vector Addition Vectors can be moved parallel to themselves without changing the resultant. the red arrow represents the resultant of the two vectors Stress that the order in which they are drawn is not important because the resultant will be the same.

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**Vector Addition Vectors can be added in any order.**

The resultant (d) is the same in each case Subtraction is simply the addition of the opposite vector.

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**Sample Resultant Calculation**

A toy car moves with a velocity of .80 m/s across a moving walkway that travels at 1.5 m/s. Find the resultant speed of the car. Use this to demonstrate the graphical method of adding vectors. Use a ruler to measure the two components and determine the scale. Then determine the size and direction of the resultant using the ruler and protractor. This would make a good practice problem for Section 2, when students learn how to add vectors using the Pythagorean theorem and trigonometry.

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3.2 Vector Operations

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What do you think? What is one disadvantage of adding vectors by the graphical method? Is there an easier way to add vectors? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally.

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Vector Operations Use a traditional x-y coordinate system as shown below on the right. The Pythagorean theorem and tangent function can be used to add vectors. More accurate and less time-consuming than the graphical method Direction means north, south, east, west, up, or down. It does not mean increasing or decreasing. So even though the temperature may be going “up,” it is really just increasing and has no direction.

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**Pythagorean Theorem and Tangent Function**

Remind students that the Pythagorean theorem can only be used with right triangles.

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**Pythagorean Theorem and Tangent Function**

We can use the inverse of the tangent function to find the angle. θ= tan-1 (opp/adj) Another way to look at our triangle d2 =Δx2 + Δy2 d Δy θ Δx

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Example An archaeologist climbs the great pyramid in Giza. The pyramid height is 136 m and width is 2.30 X 102m. What is the magnitude and direction of displacement of the archaeologist after she climbs from the bottom to the top?

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**Example Given: Unknown: Δy= 136m**

width is 2.30 X 102m for whole pyramid So, Δx = 115m Unknown: d = ?? θ= ??

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**Example d2 =Δx2 + Δy2 d = √Δx2 + Δy2 Calculate: d = √ (115)2 +(136)2**

d = 178m θ= tan-1 (opp/adj) θ= tan-1 (136/115) θ= 49.78°

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Example While following the directions on a treasure map a pirate walks 45m north then turns and walks 7.5m east. What single straight line displacement could the pirate have taken to reach the treasure?

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**Resolving Vectors Into Components**

Review these trigonometry definitions with students to prepare for the next slide (resolving vectors into components).

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**Resolving Vectors into Components**

Component: the horizontal x and vertical yparts that add up to give the actual displacement For the vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction). Assume that the angle is 35.0˚. Review the first solution with students, and then let them solve for the second component. 35°

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**Example Given: v= 95 km/h θ= 35.0° Unknown vx=??vy= ??**

Rearrange the equations sin θ= opp/ hyp opp=(sin θ) (hyp) cosθ= adj/ hyp adj= (cosθ)(hyp)

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**Example vy=(sin θ)(v) vx= (cosθ)(v) vy= (sin35°)(95) vy= 54.49 km/h**

vx = km/h

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Example How fast must a truck travel to stay beneath an airplane that is moving 105 km/h at an angle of 25° to the ground?

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3.3 Projectile Motion

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What do you think? Suppose two coins fall off of a table simultaneously. One coin falls straight downward. The other coin slides off the table horizontally and lands several meters from the base of the table. Which coin will strike the floor first? Explain your reasoning. Would your answer change if the second coin was moving so fast that it landed 50 m from the base of the table? Why or why not? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Many students will respond that the coin falling straight down will strike first because it travels less distance (or for some other reason). After students answer the questions, you may wish to show them the following demonstration with two quarters. (Practice a few times before class.) Place one quarter so that it extends halfway over the edge of a table, and then place the second coin next to it. Flip the second coin with your finger so it just brushes the first coin. One coin will fall nearly straight down while the other lands far out from the base of the table. Have students remain quiet so that they can hear the coins striking the floor. Students should hear them hitting at the same time, no matter how far out the second coin lands. You might extend this analogy to firing a rifle horizontally and dropping a bullet at the same time. It is hard for students to imagine that each strikes the ground at the same time (about 0.5 s after firing).

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**Projectile Motion Projectiles: objects that are launched into the air**

tennis balls, arrows, baseballs, javelin Gravity affects the motion Projectile motion: The curved path that an object follows when thrown, launched or otherwise projected near the surface of the earth Discuss the wide variety of projectiles. Tell students that the effect of air resistance is significant in many cases, but we will consider ideal examples with gravity being the only force. The effects of air were not very significant in the coin demonstration (see the Notes on the previous slide), but would be much more significant if the objects were traveling faster or had more surface area. Use the PHET web site to allow students to study projectile motion qualitatively. Go to simulations, choose “motion,” and choose then choose “projectile motion.” In this simulation, you can raise or lower the canon. Start with horizontal launches and note that the time in the air is only dependent on the height, and not on the speed of launch. You can change objects, and you can even launch a car. You also have the option of adding air resistance in varying amounts, as well as changing the launch angle. Have students determine which launch angles produce the same horizontal distance or range (complimentary angles) and find out which launch angle gives the greatest range (45°). Ask them to investigate the effect of air resistance on these results.

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**Projectile Motion Path is parabolic if air resistance is ignored**

Path is shortened under the effects of air resistance

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**Components of Projectile Motion**

As the runner launches herself (vi), she is moving in the x and y directions. Remind students that vi is the initial velocity, so it never changes. Students will learn in later slides that vx,i also does not change (there is no acceleration in the horizontal direction) but vy,idoes change (because of the acceleration due to gravity).

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Projectile Motion Projectile motion is free fall with an initial horizontal speed. Vertical and horizontal motion are independent of each other. Vertically the acceleration is constant (-10 m/s2 ) We use the 4 acceleration equations Horizontally the velocity is constant We use the constant velocity equations The 4th and 5th summary points are essential for problem solving. Emphasize these points now, and return to them as students work through problems.

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Projectile Motion Components are used to solve for vertical and horizontal quantities. Time is the same for both vertical and horizontal motion. Velocity at the peak is purely horizontal (vy= 0).

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Example The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas river. Suppose you kick a rock horizontally off the bridge at 5 m/s. How long would it take to hit the ground and what would it’s final velocity be?

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**Example Given: d = 321m a = 10m/s2 vi= 5m/s t = ?? vf = ??**

REMEMBER we need to figure out : Up and down aka free fall (use our 4 acceleration equations) Horizontal (use our constant velocity equation)

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**Classroom Practice Problem (Horizontal Launch)**

People in movies often jump from buildings into pools. If a person jumps horizontally by running straight off a rooftop from a height of 30.0 m to a pool that is 5.0 m from the building, with what initial speed must the person jump? Answer: 2.0 m/s As the students look at the equations, they will not find a single equation that allows them to solve this problem. First, as is often the case, they must solve for time using the height of the building (y) and the acceleration of gravity (ag). Then, they can use this time with the horizontal distance (x) to find the horizontal speed (vx).

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**Projectiles Launched at an Angle**

We will make a triangle and use our sin, cos, tan equations to find our answers Vy = V sin θ Vx = V cosθ tan = θ(y/x)

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**Classroom Practice Problem (Projectile Launched at an Angle)**

A golfer practices driving balls off a cliff and into the water below. The edge of the cliff is 15 m above the water. If the golf ball is launched at 51 m/s at an angle of 15°, how far does the ball travel horizontally before hitting the water? Answer: 1.7 x 102m (170 m) One option is to first solve for t in the vertical motion equations. This requires the use of the quadratic equation. Then, t can be used to find the horizontal distance in the horizontal motion equations. The problem can also be divided into two parts and solved without a quadratic equation. First, find the time required to reach the peak where vy is zero. Then, find the height reached and add it onto the 15 m. Finally, find the time required to fall from this height, and use the total time to find the horizontal distance.

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