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 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 t o = 0.0s, x o = 5.0m, v o = +2.0m/s t = 1.2s, x = 3.08m, v = -5.2m/s 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? 1.Yes 2.No

9 Acceleration (m/s/s)

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

11 Previous Example: t o = 0.0s, x o = 5.0m, v o = +2.0m/s t = 1.2s, x = 3.08m, v = -5.2m/s

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

13 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 Ex: V 2 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 + 5t 2 ) = t 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

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25 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?

26 Case Study: 100 meter track-race 1.a = const., 0-60 m 2.top speed of 16 m/s at 60 m. 3.a = 0, m

27 a) Acceleration and Time 100m Race

28 b) Time and Distance: Last 40meters of race at constant speed of 16m/s. Race Time = t I + t II = 7.5s + 2.5s = 10.0s 100m Race

29 v = v o + at. 16 = 0 + a(7.5) a = 16/7.5 = 2.13 m/s 2. c) We can also use time found in part (a) in velocity equation to get the acceleration of the runner in 1 st part of the race.  x = v avg t = {(v o + v)/2}t = {(0 + 16)/2)}(7.5) = (8)(7.5) = 60m. d) Distance using v avg

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31 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: Using v(t) equation: Consistent answer: How long did it take the object to reach v = 0?

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33 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: 1.speeds up all the time. 2.slows down all the time. 3.speeds up part of the time and slows down part of the time. 4.moves at a constant velocity. time position

34 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v v(t) a +- Motion Diagram Example

35 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 Net Force, Acceleration, & Motion Diagrams

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

37 Vehicle Average Stopping Distance at 55 mph (includes reaction time) Passenger car190 ft. Tractor-trailer (loaded) with cool brakes 256 ft. Tractor-trailer (loaded) with hot brakes 430 ft. Tractor-trailer (empty)249 ft. Tractor only (bobtail)243 ft.

38 Vehicle Stopping Distance from 60 mi/hr Accel. feetmetersft/s 2 m/s 2 BMW M Dodge Colt GL

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40 Time to Stop BMW Colt

41 y and v graphs for tossed object in “free-fall”

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