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- 1. Two-Dimensional Motion and Vectors<br />Chapter 3 pg. 81-105<br />
- 2. What do you think?<br />How are measurements such as mass and volume different from measurements such as velocity and acceleration?<br />How can you add two velocities that are in different directions?<br />
- 3. Projectile motion can be described by the horizontal and vertical components of motion.<br />
- 4. Introduction to Vectors<br />Scalar - a quantity that has magnitude but no direction<br />Examples: volume, mass, temperature, speed<br />Vector - a quantity that has both magnitude and direction<br />Examples: acceleration, velocity, displacement, force<br />
- 5. Vector Properties<br />Vectors are generally drawn as arrows.<br />Length represents the magnitude<br />Arrow shows the direction<br />Resultant - the sum of two or more vectors<br />Make sure when adding vectors that<br />You use the same unit<br />Describing similar quantities<br />
- 6. Components of Vectors<br />A ball’s velocity can be placed into into horizontal and vertical components.<br />
- 7. Vector Addition<br />Vectors can be moved parallel to themselves without changing the resultant. <br />the red arrow represents the resultant of the two vectors<br />
- 8. Vector Addition<br />Vectors can be added in any order.<br />The resultant (d) is the same in each case<br />Subtraction is simply the addition of the opposite vector.<br />
- 9. 3.2 Vector Operations<br />
- 10. What do you think?<br />What is one disadvantage of adding vectors by the graphical method?<br />Is there an easier way to add vectors?<br />
- 11. Vector Operations<br />Use a traditional x-y coordinate system as shown below on the right.<br />The Pythagorean theorem and tangent function can be used to add vectors.<br />More accurate and less time-consuming than the graphical method<br />
- 12. Pythagorean Theorem and Tangent Function<br />
- 13. Pythagorean Theorem and Tangent Function<br />We can use the inverse of the tangent function to find the angle.<br />θ= tan-1 (opp/adj)<br />Another way to look at our triangle<br />d2 =Δx2 + Δy2<br />d<br />Δy<br />θ<br />Δx<br />
- 14. Example<br />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?<br />
- 15. Example<br />Given:<br />Δy= 136m <br />width is 2.30 X 102m for whole pyramid<br />So, Δx = 115m<br />Unknown:<br />d = ?? θ= ??<br />
- 16. Example<br />Calculate:<br />d2 =Δx2 + Δy2<br />d = √Δx2 + Δy2<br />d = √ (115)2 +(136)2<br />d = 178m<br />θ= tan-1 (opp/adj)<br />θ= tan-1 (136/115)<br />θ= 49.78°<br />
- 17. Example<br />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?<br />
- 18. Resolving Vectors Into Components<br />
- 19. Resolving Vectors into Components<br />Component: the horizontal x and vertical yparts that add up to give the actual displacement<br />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 that the angle is 35.0˚.<br />35°<br />
- 20. Example<br />Given: v= 95 km/h θ= 35.0°<br />Unknown vx=??vy= ??<br />Rearrange the equations <br />sin θ= opp/ hyp or sin θ=vy/v<br />opp=(sin θ) (hyp) or vy=(sin θ)(v)<br />cosθ= adj/ hyp or cosθ= vx/v<br />adj= (cosθ)(hyp) or vx= (cosθ)(v)<br />
- 21. Example<br />vy=(sin θ)(v)<br />vy= (sin35°)(95)<br />vy= 54.49 km/h<br />vx= (cosθ)(v)<br />vx = (cos 35°)(95)<br />vx = 77.82 km/h<br />
- 22. Example <br />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?<br />
- 23. 3.3 Projectile Motion<br />
- 24. What do you think?<br />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. <br />Which coin will strike the floor first?<br />Explain your reasoning.<br />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?<br />
- 25. Projectile Motion<br />Projectiles: objects that are launched into the air<br />tennis balls, arrows, baseballs, javelin <br />Gravity affects the motion<br />Projectile motion:<br />The curved path that an object follows when thrown, launched or otherwise projected near the surface of the earth<br />
- 26. Projectile Motion<br />Path is parabolic if air resistance is ignored<br />Path is shortened under the effects of air resistance<br />
- 27. 5.4 Projectile Motion<br />Projectile motion can be separated into components. <br />Roll a ball along a horizontal surface, and its velocity is constant because no component of gravitational force acts horizontally. <br />
- 28. 5.4 Projectile Motion<br />Projectile motion can be separated into components. <br />Roll a ball along a horizontal surface, and its velocity is constant because no component of gravitational force acts horizontally. <br />Drop it, and it accelerates downward and covers a greater vertical distance each second.<br />
- 29. Components of Projectile Motion<br />As the runner launches herself (vi), she is moving in the x and y directions.<br />
- 30. Projectile Motion<br />Projectile motion is free fall with an initial horizontal speed.<br />Vertical and horizontal motion are independent of each other.<br />Vertically the acceleration is constant (10 m/s2 )<br />We use the 4 acceleration equations<br />Horizontally the velocity is constant<br />We use the constant velocity equations<br />
- 31. Projectile Motion<br />Components are used to solve for vertical and horizontal quantities.<br />Time is the same for both vertical and horizontal motion.<br />Velocity at the peak is purely horizontal (vy= 0).<br />
- 32. Example<br />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?<br />
- 33. Example<br />Given: d = 321m a = 10m/s2<br />vi= 5m/s t = ?? vf = ??<br />REMEMBER we need to figure out :<br />Up and down aka free fall (use our 4 acceleration equations)<br />Horizontal (use our constant velocity equation)<br />
- 34. Classroom Practice Problem (Horizontal Launch)<br />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?<br />Answer: 2.0 m/s<br />
- 35. Projectiles Launched at an Angle<br />We will make a triangle and use our sin, cos, tan equations to find our answers<br />Vy = V sin θ<br />Vx = V cosθ<br />tan = θ(y/x)<br />
- 36. Classroom Practice Problem(Projectile Launched at an Angle)<br />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?<br />Answer: 1.7 x 102m (170 m)<br />

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