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Kinematics… Motion in a straight line

Kinematics… Motion in a straight line. 1-2 Displacement vs Distance Average Velocity 1-4 Acceleration 1.5 finding the motion of an object. Motion in a straight line. Mechanics = Study of objects in motion. Position ( x ) – where you are located

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Kinematics… Motion in a straight line

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  1. Kinematics…Motion in a straight line 1-2 Displacement vs Distance Average Velocity 1-4 Acceleration 1.5 finding the motion of an object

  2. Motion in a straight line • Mechanics = Study of objects in motion. • Position (x) – where you are located • Distance (d) – how far you have traveled, regardless of direction • Displacement (x) – where you are in relation to where you started • 2 parts to mechanics. • Kinematics =Description of HOWobjects move. • Dynamics =WHYobjects move. Norah Ali Al- Moneef

  3. 1-2 Displacement vs DistanceAverage Velocity • Displacement is a vector that points from an object’s initial position to its final position • and has a magnitude that equals the shortest distance between the two positions. _Only depends on the initial and final positions • Independent of actual paths between the initial and final positions • Distance is a scalar • Depends on the initial and final positions as well as the actual path between them Norah Ali Al- Moneef

  4. Displacement The displacementΔx is a vector that points from the initial position to the final position. SI Unit of Displacement: meter (m) Norah Ali Al- Moneef

  5. LAKE A B Comparing Vector & Scalar Values Displacement (a vector) versus distance (a scalar) stop start We want to get from point A to point B. If we follow the road around the lake our direction is always changing. There is no specific direction. The distance traveled on the road is a scalar quantity. A straight line between A and B is the displacement. It has a specific direction and is therefore a vector. Norah Ali Al- Moneef

  6. Displacement This type of x(t) plot shows the position of an object at any time, e.g., x (m) Position at t=3 s, x(3) = 1 m 3 t (s) 4 Displacement between t=1 s and t=5 s Dx = 1.0 m - 2.0 m = -1.0 m -3 Norah Ali Al- Moneef

  7. Given the train’s initial position and its final position what is the displacement of the train? What is the distance traveled by the train ? Displacement = Norah Ali Al- Moneef

  8. Example: A boy travels from D to A,A to B .B to C.C to D Displacement from D to D ( which are initial and final points ) = 0 Distance traveled = 8 +4+8+4 = 24 m Norah Ali Al- Moneef

  9. Example: A student on her way to school walks four blocks south, five blocks west, and another four blocks south, as shown in the diagram. What is the distance she walks ? What is displacement from home to school ? Distance = 4 blocks + 5 blocks +4 blocks = 13 blocks Displacement = ( 82 + 5 2 ) ½ = 9.43 blocks Norah Ali Al- Moneef

  10. 3 tan  = 4 3 km  4 km Example: It is easy to adddisplacements if they are perpendicular to each other. E.g. d 2 = 42 + 32 d = 5 km N d 5 km  = 37 Norah Ali Al- Moneef

  11. Example : Distance = 4 m + 3 m =7 m Displacement = 5 m Norah Ali Al- Moneef

  12. N 3 km 7 km north 4 km Example: A car travels 4 km north then 3 km north total distance = (3 + 4) km = 7 km total displacement = (3 + 4) kmnorth = 7 kmnorth Norah Ali Al- Moneef

  13. N 1 km north 3 km 4 km • example A car travels 4 km north then 3 km south total distance = = (3 + 4) km = 7 km total displacement = = (3 + 4) kmnorth = 1 kmnorth Norah Ali Al- Moneef

  14. 4km 4km 5km 5km Example N A car travels 5 km north and 4 km east. (a) Total distance travelled = ? Total distance = 5 + 4 = 9km (b) What is the displacement? = 6.40 km displacement = tan  =   = 38.7  Total displacement: 6.40 km 38.7° east of north 6.4km  Norah Ali Al- Moneef

  15. example A ball hung by a string swings from X to Y. What is the size of the displacement ofthe ball? 60o 1 m A p/3 m B 1 m C 1 m towardsthe right 1 m X Y Norah Ali Al- Moneef

  16. 70 m Example A girl cycles a circulartrack of diameter 70 m and stops atthe starting point. (a)Distance travelled = ? (b)What is her displacement? Distance travelled = perimeter of track = p× 70 = 220 m displacement = 0 m Norah Ali Al- Moneef

  17. Example: Starting from origin, O a person walks 90-m east, then turns around and walks 40-m west. What is the total walked distance? What is the displacement? 130 m A: 50-m, due east. Norah Ali Al- Moneef

  18. Speed Speed can be defined in a couple of ways: How fast something is moving The distance covered in a certain amount of time The rate of change of the position of an object Units for speed are: miles / hour (mi/hr) kilometers / hour (km/hr) feet / second (ft/s) This is the standard unitmeters / second (m/s) The speed has no direction and is always expressed as a positive number Norah Ali Al- Moneef

  19. Speed and Velocity The average speed being the distance traveled divided by the time required to cover the distance: How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m /s? Distance = 5400 s x 2.22 m / s = 11988 m Norah Ali Al- Moneef

  20. No yes Norah Ali Al- Moneef

  21. Graphing Skills • Motion can be studied using a distance vs. time graph. • time (x-axis) = independent variable • distance (y-axis) = dependent variable • The slope of a distance vs. time graph equals speed. Calculating Slope The slope of a straight line equals the vertical change divided by the horizontal change. Determine the slope of the blue line shown in the distance vs. time graph. Norah Ali Al- Moneef

  22. Graphing Skills, continued 1. Choose two points that you will use to calculate the slope. Point 1: t = 1 s and d = 6 m Point 2: t = 4 s and d = 12 m • Calculate the vertical change and the horizontal change. vertical change = 12 m – 6 m = 6 m horizontal change = 4 s – 1 s = 3 s • Divide the vertical change by the horizontal change. slope = 6 m /3 s = 2 m/s Norah Ali Al- Moneef

  23. Average velocity Average velocity • Can be positive or negative • Depends only on initial/final positions • e.g., if you return to original position, average velocity is zero Speed and velocity are not the same. Velocity requires a directional component and is therefore a vector quantity. Speed tells us how fast we are going but not which way. Speed is a scalar (direction doesn’t count!) Norah Ali Al- Moneef

  24. example Answer is d Norah Ali Al- Moneef

  25. Graphical Representation of Average Velocity Between A and D , v is slope of blue line Norah Ali Al- Moneef

  26. 2 km h–1 Example A man walks from A to B at 1 km h–1, and returns at 2 km h–1. whole journey = 2 km Suppose AB = 1 km 1 kmh–1 A B what is the average speed for thewhole trip ? Time for whole trip = = 1 h + 0.5 h = 1.5 h Ave. speed = distance / time 2/1.5 =1.33 Km / s Norah Ali Al- Moneef

  27. (7 + 3) km = (10/60) h Example A car travels 7 km north and then 3 km west in 10 minutes. Find (a) average speed, Ave. speed = 3 km B C distance travelled time taken 7 km = 60 kmh–1 A Norah Ali Al- Moneef

  28. Example A car travels 7 km north and then 3 km west in 10 minutes. Find (b) ave. velocity? 3 km B C AC = 7 km = 7.62 km  q =23.2o tan q = 3/7 A Norah Ali Al- Moneef

  29. Example A car travels 7 km north and then 3 km west in 10 minutes. Find (b) ave. velocity? 3 km AC = 7.62 km, q =23.2o B C Size of ave. velocity = displacement 7.62 km 7 km = time (10/60) h = 45.7 km h–1 Ave. velocity is 45.7 km h–1, 23.2°north of west. A Norah Ali Al- Moneef

  30. example Norah Ali Al- Moneef

  31. example What is Norah Ali Al- Moneef

  32. example From A to B What is B A Norah Ali Al- Moneef

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  35. example Norah Ali Al- Moneef

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  38. Example Carol starts at a position x(t=0) = 1.5 m. At t=2.0 s, Carol’s position is x(t=2 s)=4.5 m At t=4.0 s, Carol’s position is x(t=4 s)=-2.5 m • What is Carol’s average velocity between t=0 and t=2 s? • What is Carol’s average velocity between t=2 and t=4 s? • What is Carol’s average velocity between t=0 and t=4 s? a) 1.5 m/s b) -3.5 m/s c) -1.0 m/s Norah Ali Al- Moneef

  39. Instantaneous velocity The velocity atany instant is calledinstantaneous velocity. Let time interval approach zero • Defined for every instance in time • Equals average velocity if v = constant • SPEED is absolute value of velocity If a car moves at a constant velocity... its average and instantaneous velocities have the same value. Norah Ali Al- Moneef

  40. INSTANTANEOUS VELOCITY Norah Ali Al- Moneef

  41. Graphical Representation of Instantaneous Velocity v(t=3.0) is slope of tangent (green line) Norah Ali Al- Moneef

  42. example A particle moves along a straight line such that its position is defined by s = (t3 – 3 t2 + 2 ) m. Determine the velocity of the particle when t = 4 s. At t = 4 s, the velocity = 3 (4)2 – 6(4) = 24 m/s Norah Ali Al- Moneef

  43. 1 - 4 Acceleration Acceleration:is a rate at which a velocity is changing. Instantaneous acceleration = dv / dt = d2 x / d2 t Norah Ali Al- Moneef

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  45. Acceleration is a change in velocity or a change in direction of velocity. . First car is accelerating because its velocity is increasing. Second car is accelerating because its direction is changing. Third car is accelerating because its velocity and direction are changing. Norah Ali Al- Moneef

  46. Positive/Negative Acceleration Positive Acceleration –produces an increase in speed in the positive direction, or decrease of speed in the negative direction. Negative Acceleration – produces a decrease in speed in the positive direction or an increase of speed in the negative direction. Norah Ali Al- Moneef

  47. Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the directions of velocity and acceleration : Norah Ali Al- Moneef

  48. Acceleration There is a difference between negative acceleration and deceleration: Negative acceleration is acceleration in the negative direction as defined by the coordinate system. Deceleration occurs when the acceleration is opposite in direction to the velocity. Norah Ali Al- Moneef

  49. Example A car’s velocity at the top of a hill is 10 m/s. Two seconds later it reaches the bottom of the hill with a velocity of 26 m/s. What is the acceleration of the car? The car is increasing its velocity by 8 m/s for every second it is moving. Norah Ali Al- Moneef

  50. example Norah Ali Al- Moneef

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