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# Graphical analysis of motion

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• An object at zero velocity is at rest. The object will not change position.
• Positive velocity indicates direction towards the RIGHT or EAST
• Straight line upward indicates positive constantvelocity
• Negative velocity indicates an object moving toward the LEFT or WEST
• A straight downward slope indicates constant negative velocity to which an object moves constantly leftward
• Positive velocity indicates object moves towards RIGHT, Positive acceleration indicates object speeds up
• Increasing positive slope shows acceleration.. A change in velocity speeding up
• ### Graphical analysis of motion

1. 1. Tabulating Time, Displacement and Velocity at a given Acceleration<br />
2. 2. A car starting from rest, accelerating at 2m/s2 to the right? Describe its position every second from 0-5 sec.<br />Since the starts at rest. vo= 0m/s, deriving vf=at<br />Using d=vavet<br />t=0, v=0, d=0 at 1<br />At 1 sec<br />vf=2m/s2(1s) = 2m/s<br />d=2m/s(1s)=2m<br />
3. 3. Graphical Analysis of Motion<br />
4. 4. Position –Time Graph<br />
5. 5. Case 1: Zero Velocity<br />50<br />1<br />2<br />3<br />4<br />5<br />40<br />30<br />Position<br />20<br />10<br />0<br />Time<br />Object at 30 meters with zero velocity<br />Object at rest and in the same position for a long time<br />
6. 6. 30<br />55<br />10<br />15<br /> <br /> <br />20<br />25<br />35<br />40<br /> <br /> <br /> 45<br />50<br />t=1<br />t=3<br />t=4<br />t=5<br />t=2<br />Case 2: Positive, Constant velocity<br />Consider a car moving with a constant velocity of 10 m/s to the right<br />
7. 7. 50<br />1<br />2<br />3<br />4<br />5<br />40<br />30<br />Position<br />20<br />10<br />0<br />Time<br />Straight line sloping upward<br />Case 2: Positive, Constant velocity<br />
8. 8. 30<br />55<br />10<br />15<br /> <br /> <br />20<br />25<br />35<br />40<br /> <br /> <br /> 45<br />50<br />t=4<br />t=1<br />t=2<br />t=3<br />t=5<br />t=0<br />60<br />Case 3: Negative, Constant velocity<br />Consider a car moving with a constant velocity of 10 m/s to the left<br />
9. 9. 50<br />1<br />2<br />3<br />4<br />5<br />40<br />30<br />Position<br />20<br />10<br />0<br />Time<br />Straight line sloping downward<br />Case 3: Negative, Constant velocity<br />
10. 10. t=4<br />t=1<br />t=0<br />t=3<br />t=2<br />40<br />2.5<br />0<br /> 10<br /> <br /> <br />22.50<br />Case 4: Positive Velocity, Positive Acceleration<br />Consider a car starts from rest and accelerates at 5m/s2<br />
11. 11. 50<br />1<br />2<br />3<br />4<br />5<br />40<br />30<br />Position<br />20<br />10<br />0<br />Time<br />Case 4: Positive Velocity, Positive Acceleration<br />Increasing, positive slope<br />The car is accelerating or speeding up to the right<br />
12. 12. t=0<br />t=2<br />t=1<br />t=3<br />t=4<br />t=5<br />0<br />40<br />62.25 <br /> <br />22.50<br />60<br />52.25<br />Case 5: Positive Velocity, Negative Acceleration<br />Consider a car moving at 25m/s and accelerates at -5m/s2<br />
13. 13. 1<br />2<br />3<br />4<br />5<br />50<br />40<br />Position<br />30<br />20<br />10<br />60<br />0<br />Time<br />Case 5: Positive Velocity, Negative Acceleration<br />Decreasing Positive slope<br />The car is decelerating or slowing down to the right<br />
14. 14. t=5<br />t=1<br />t=2<br />t=3<br />t=4<br />t=0<br />0<br />40<br />62.25 <br /> <br />22.50<br />60<br />52.25<br />Case 6: Negative Velocity, Negative Acceleration<br />Consider a car at position 62.25m starting from rest and moving leftward with an acceleration of -5m/s2<br />
15. 15. Increasing Negative slope<br />Accelerating or speeding up to the left<br />1<br />2<br />3<br />4<br />5<br />50<br />40<br />Position<br />30<br />20<br />60<br />10<br />0<br />Time<br />Case 6: Negative Velocity, Negative Acceleration<br />
16. 16. t=5<br />t=2<br />t=3<br />t=4<br />t=1<br />t=0<br />62.5<br />40<br />2.5<br />0<br /> 10<br /> <br /> <br />22.50<br />Case 7: Negative Velocity, Positive Acceleration<br />Consider a car at position 62.25m starting at -25m/s but accelerating at +5m/s2<br />
17. 17. 1<br />2<br />3<br />4<br />5<br />50<br />40<br />Position<br />30<br />20<br />60<br />10<br />0<br />Time<br />Case 7: Negative Velocity, Positive Acceleration<br />Decreasing, negative slope<br />The car is decelerating or slowing down to the left<br />
18. 18. Velocity – Time Graph<br />
19. 19. Case 1: Zero Velocity<br />Object at rest in same position<br />V+<br />0<br />V-<br />
20. 20. V+<br />0<br />V-<br />Case 2: Positive Constant Velocity<br />Object is moving at constant velocity towards the right<br />
21. 21. V+<br />0<br />V-<br />Case 3: Negative constant Velocity<br />Object is at constant negative velocity indicating motions towards the LEFT<br />
22. 22. V+<br />0<br />V-<br />Case 4: Positive velocity, Positive acceleration<br />Indicates the object speeds up towards the right<br />
23. 23. V+<br />0<br />V-<br />Case 5: Positive velocity, Negative acceleration<br />Indicates the object slows down towards the right<br />
24. 24. V+<br />0<br />V-<br />Case 6: Negative velocity, Negative acceleration<br />Indicates the object speeds up towards the left<br />
25. 25. V+<br />0<br />V-<br />Case 4: Negative velocity, Positive acceleration<br />Indicates the object slows down towards the left<br />
26. 26. Fill in the blanks by describing position time graphs and velocity time graphs. <br />In a v-t graph an object that moves with a positive velocity and a negative acceleration; the object slows down to the ________________.<br /> <br />In a v-t graph an object that has a constant negative velocity is represented by a straight line parallel but _______ the x-axis. <br /> <br />In a p-t graph an object that is moving at a negative acceleration, negative velocity has an increasing negative slope which __________towards the left.<br /> <br />
27. 27. A car that moves to the right and speeding up both has ________ velocity and acceleration. An object that is in zero velocity is at 20 meter; its position is constant and is represented by a line which is ____________ to the x-axis.<br />
28. 28. Case 4: sloping upward above x<br />Case 5: sloping down above x<br />Case 6: sloping down below x-axis<br />Case7: sloping up below x-axis<br />
29. 29. One dimensional vertical motion<br />Free fall: a state where an object which moves under the sole influence of GRAVITY. (acceleration due to gravity)<br />
30. 30. Two important characteristics of a free falling object:<br />Gravity = 9.8m/s2<br />Downward direction<br />
31. 31. Mass and Acceleration<br />Acceleration is directly proportional to the force and inversely proportional to the mass.<br />
32. 32. When will an object fall faster?<br />An object will fall faster if there is an appreciable amount of air resistance.<br />Air resistance is the friction present in the air. The larger the velocity of an object the larger is the air resistance.<br />
33. 33. Quantities involved in free fall<br />vo= velocity initialvf= final velocityg= gravity (9.8 m/s2)t=timey= displacement<br />
34. 34. Equations of one-dimensional vertical motion<br />vf=v0 + gty= v0t + 1/2 gt2vf2=vo2 + 2gy<br />
35. 35. Case 1 : Dropped (vo=0)Objects that are dropped has an initial velocity of ZERO.Example:A ball is dropped from a height of 50m Compute for the time of flight to reach the ground.<br />50 meters<br />
36. 36. From y= v0t + 1/2 gt2 and vo=0 <br /> t= 2𝑦𝑔<br />Same as velocity before impact from vf2=vo2 + 2gy<br />vf=2𝑔𝑦<br /> <br />
37. 37. Solving for time and velocity of an object dropped at 50 meters above the ground.<br />t= 2𝑦𝑔 t= 2(50𝑚)9.8𝑚/𝑠2 = 3.2 s<br />vf=2𝑔𝑦=29.8𝑚𝑠250𝑚<br />= 31.3 m/s<br /> <br />
38. 38. Creating the time-velocity-displacement table of the motion, from 0s to 3.2 sec intervals of 0.8 s we use the following formulas :<br />vf=v0+ gty= v0t + 1/2 gt2<br />50 meters<br />