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Topics. Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals. Average Speed. distance: total path length speed: rate of travel (e.g. 50 mph) Average Speed: distance/time (e.g. 100m in 3.0s). 0.

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  1. Topics • Distance, Location, Speed • Speed and Direction • Directional quantities • Acceleration • Free Fall • Graphs of Motion • Derivatives and Integrals

  2. Average Speed • distance: total path length • speed: rate of travel (e.g. 50 mph) • Average Speed: distance/time (e.g. 100m in 3.0s)

  3. 0 Displacement: Change in Position SI Unit: meters (m)

  4. Velocity (m/s)

  5. Velocity Examples • average velocity: 60mph toward Dallas • instantaneous velocity: 11:47am: Northbound, 83mph

  6. Example: Average Velocity to = 0.0s, xo = 5.0m, vo = +2.0m/s t = 1.2s, x = 3.08m, v = -5.2m/s 0 Note that velocities always have directional information. Here the “-” sign means –x direction.

  7. Scalars & Vectors • Scalar: size only • e.g. speed, distance, time • Vector: magnitude and direction • e.g. displacement, velocity, acceleration

  8. A honeybee travels 2 km round trip before returning. Is the displacement for the trip the same as the distance traveled? • Yes • No

  9. Acceleration (m/s/s)

  10. 0 Example: Car goes from 10m/s to 15m/s in a time of 2.0 seconds. Calculate the average acceleration.

  11. Previous Example: to = 0.0s, xo = 5.0m, vo = +2.0m/s t = 1.2s, x = 3.08m, v = -5.2m/s 0

  12. 0 Motion Diagrams • velocity arrow and position • zero velocity is a “dot” • acceleration & net-force directions: parallel to Dv • Example: slowing, reversing direction

  13. 0 Kinematic Equations of Constant Acceleration

  14. Displacement and x vs. t Graph

  15. x vs. t Graph • slope is velocity

  16. v vs. t Graph • slope is acceleration

  17. Human Acceleration In the 1988 Olympics, Carl Lewis reached the 20m mark in 2.96s. Calculate average acceleration.

  18. Cheetah Acceleration A cheetah can accelerate from 0 to 20m/s in 2.0s. What is the average acceleration?

  19. 0 Ex: V2 Equation Approximate Stopping Accelerations in m/s/s: Dry Road: ~ 9 (anti-lock) ~ 7 (skidding) Wet Road: ~ 4 (anti-lock) ~ 2 (skidding) At 60mph = 27m/s, what is the stopping distance of a skid on a wet road?

  20. Free-Fall • only gravity acts • air-friction is negligible • a = 9.8m/s/s downward

  21. Calculus of Linear Motion • derivatives and integrals • Examples: • dx/dt = v dv/dt = a • d/dt(3 + 4t + 5t2) = 4 + 10t • v = integral of acceleration

  22. Velocity Example:

  23. Summary: • speed: rate of travel • average speed: distance/time. • displacement: change in position • velocity: rate position changes • acceleration: rate velocity changes • kinematic equation set • free fall: constant acceleration. • graphs and slopes • derivatives and integrals of polynomials

  24. 0 Example: A solid metal ball is projected directly upward with velocity +5.0m/s. How high does it go? How long does it take to return to same height?

  25. Case Study: 100 meter track-race 0 • a = const., 0-60 m • top speed of 16 m/s at 60 m. • a = 0, 60-100 m

  26. 0 100m Race a) Acceleration and Time

  27. 0 100m Race b) Time and Distance: Last 40meters of race at constant speed of 16m/s. Race Time = tI + tII = 7.5s + 2.5s = 10.0s

  28. 0 c) We can also use time found in part (a) in velocity equation to get the acceleration of the runner in 1st part of the race. v = vo + at. 16 = 0 + a(7.5) a = 16/7.5 = 2.13 m/s2. d) Distance using vavg Dx = vavgt = {(vo + v)/2}t = {(0 + 16)/2)}(7.5) = (8)(7.5) = 60m.

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  30. Using v(t) equation: 0 Example: An object has velocity of +2.0m/s at x = 5.0m and at t = 0.0s. At t = 1.2s it has velocity of -5.2m/s and position x = 3.08m. Average Acceleration: Consistent answer: How long did it take the object to reach v = 0?

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  32. A train moves along a straight track. The graph shows the position as a function of time for this train. Note that the speed at an instant is the slope of the line at any point on the line. The graph shows that the train: • speeds up all the time. • slows down all the time. • speeds up part of the time and slowsdown part of the time. • moves at a constant velocity. position time

  33. Motion Diagram Example 0 A car travels West at 20m/s. It begins to slow. Use the convention that East is +x. The acceleration of the car is considered positive since if it slowed to 19m/s in 1.0s, then Motion Diagram: v(t) Dv a - +

  34. Net Force, Acceleration, & Motion Diagrams 0 Example: A car starts from rest and travels West with uniformly increasing speed. Use the convention that East is +x. Is the acceleration + or -? Is the total force acting on the car + or -? Draw a motion diagram. Assume it goes from 0 to -10m/s in 10s. Net-force parallel to acceleration, i.e. force is – direction. motion diagram

  35. Example using Acceleration 0 A car can accelerate at 6m/s/s. The time to go from 40mph to 60mph is:

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  39. 0 Time to Stop BMW Colt

  40. y and v graphs for tossed object in “free-fall” 0

  41. Realistic Car? 0 Determine how realistic 6m/s/s is for a car by computing the 0 to 60mph time: Good time, but can be done.

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