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Motion Along a Straight Line

2. Motion Along a Straight Line. Chapter 2: Motion Along a Line. Position & Displacement Speed & Velocity Acceleration Describing motion in 1D Free Fall. 2. Motion. Motion is divided into two areas of study: Kinematics This will be our focus in chapter 2.

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Motion Along a Straight Line

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  1. 2 Motion Along a Straight Line

  2. Chapter 2: Motion Along a Line • Position & Displacement • Speed & Velocity • Acceleration • Describing motion in 1D • Free Fall 2

  3. Motion • Motion is divided into two areas of study: • Kinematics • This will be our focus in chapter 2. • Kinematics describes the movement of the object. • Dynamics • Will come in Chapter 4 and after. • Dynamics answers the “Why is this object moving?” question.

  4. One Dimensional Motion Kinematics

  5. Position and Displacement • Position: coordinate of position – distance to the origin (m) P O x x

  6. x (cm) 1 2 0 1 2 Example Position The position of the ball is The + indicates the direction to the right of the origin.

  7. Displacement: change in position x0 = original (initial) location x = final location Δx = x - x0 Example: x0 = 1m, x = 4m, Δx = 4m – 1m = 3m Displacement Δx 1m 4m

  8. Displacement: change in position x0 = original (initial) location x = final location Δx = x - x0 Example: x0 = 4m, x = 1 m, Δx = 1m – 4m = - 3m Displacement Δx 1m 4m

  9. Distance traveled usually different from displacement. Distance always positive. Previous example: always 3 m. Displacement, Distance

  10. Distance and Speed: Scalar Quantities Distance is the path length traveled from one location to another. It will vary depending on the path. Distance is a scalar quantity—it is described only by a magnitude.

  11. a) 3 m b) 6 m c) -3 m d) 0 m What is the displacement in the situation depicted bellow? 1m 4m

  12. a) 3 m b) 6 m c) -3 m d) 0 m What is the distance traveled in the situation depicted bellow? 1m 4m

  13. Position, Displacement

  14. One-Dimensional Displacement and Velocity: Vector Quantities Displacement is a vector that points from the initial position to the final position of an object.

  15. Velocity: Rate of Change of Position • Average velocity = displacement/time interval vav = (x-xo)/(t-to)=Δx/Δt • displacement = average velocity x time interval Δx =vavΔt • Position x. Since Δx = x – xo: x = xo + vavΔt (Equation of a straight line) Note that an object’s position coordinate may be negative, while its velocity may be positive; the two are independent.

  16. Example

  17. Velocity and Position • Average velocity = displacement/time interval • In general, the average velocity is the slope of the line segment that connects the positions at the beginning and end of the time interval Section 2.2

  18. Constant Velocity velocity = slope of graph position vs. time v = 8m/2s = 4 m/s

  19. Graphs of Velocity Different ways of visualizing uniform velocity:

  20. Instantaneous Velocity • Average velocity doesn’t tell us anything about details during the time interval • To look at some of the details, smaller time intervals are needed • The slope of the curve at the time of interest will give the instantaneous velocity at that time • Will be referred to as velocity in the text Section 2.2

  21. Instantaneous Velocity Instantaneous Velocity = Slope of the tangent line Example v = 5.2 m/s

  22. One-Dimensional Displacement and Velocity: Vector Quantities This object’s velocity is not uniform. Does it ever change direction, or is it just slowing down and speeding up?

  23. Average Speed Average speed is the total distance traveled divided by the elapsed time: • Average Speed = Distance/Time • Distance = Average Speed x Time

  24. Distance and Speed: Scalar Quantities Since distance is a scalar, speed is also a scalar (as is time). Instantaneous speed is the speed measured over a very short time span. This is what a speedometer reads.

  25. One-Dimensional Displacement and Velocity: Vector Quantities For motion in a straight line with no reversals, the average speed and the average velocity are the same. Otherwise, they are not; indeed, the average velocity of a round trip is zero, as the total displacement is zero!

  26. Example Usein Bolt crossed the 100 m dash race in 9.69 s, ahead of the second athlete by 15 m. How long did it take for the second athlete to complete this race? Known: distance = 100 m, time = 9.69 s, Δd = 15 m Find: time of second athlete. distance traveled by second athlete in 9.69 s = 100 m – 15 m = 85 m speed of second athlete = 85m/9.69s = 8.77 m/s time = distance/speed = 100m/8.77m/s = 11.4 s

  27. ConcepTest 2.1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? 1) yes 2) no

  28. ConcepTest 2.1 Walking the Dog You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? 1) yes 2) no Yes, you have the same displacement. Since you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement. Follow-up: Have you and your dog traveled the same distance?

  29. ConcepTest 2.2 Displacement 1) yes 2) no 3) it depends on the coordinate system Does the displacement of an object depend on the specific location of the origin of the coordinate system?

  30. 10 20 30 40 50 30 40 50 60 70 ConcepTest 2.2 Displacement 1) yes 2) no 3) it depends on the coordinate system Does the displacement of an object depend on the specific location of the origin of the coordinate system? Since the displacement is the difference between two coordinates, the origin does not matter.

  31. ConcepTest 2.3 Position and Speed 1) yes 2) no 3) it depends on the position If the position of a car is zero, does its speed have to be zero?

  32. ConcepTest 2.3 Position and Speed 1) yes 2) no 3) it depends on the position If the position of a car is zero, does its speed have to be zero? No, the speed does not depend on position; it depends on the change of position. Since we know that the displacement does not depend on the origin of the coordinate system, an object can easily start at x = –3 and be moving by the time it gets to x = 0.

  33. ConcepTest 2.4 Odometer 1) distance 2) displacement 3) both Does the odometer in a car measure distance or displacement?

  34. ConcepTest 2.4 Odometer 1) distance 2) displacement 3) both Does the odometer in a car measure distance or displacement? If you go on a long trip and then return home, your odometer does not measure zero, but it records the total miles that you traveled.That means the odometer records distance. Follow-up: How would you measure displacement in your car?

  35. ConcepTest 2.5 Speedometer 1) velocity 2) speed 3) both 4) neither Does the speedometer in a car measure velocity or speed?

  36. ConcepTest 2.5 Speedometer 1) velocity 2) speed 3) both 4) neither Does the speedometer in a car measure velocity or speed? The speedometer clearly measures speed, not velocity. Velocity is a vector (depends on direction), but the speedometer does not care what direction you are traveling. It only measures the magnitude of the velocity, which is the speed. Follow-up: How would you measure velocity in your car?

  37. ConcepTest 2.6a Cruising Along I 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip?

  38. ConcepTest 2.6a Cruising Along I 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip? It is 40 mi/hr in this case. Since the average speed is distance/time and you spend the same amount of time at each speed, then your average speed would indeed be 40 mi/hr.

  39. ConcepTest 2.6b Cruising Along II 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip?

  40. ConcepTest 2.6b Cruising Along II 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip? It is not 40 mi/hr! Remember that the average speed is distance/time. Since it takes longer to cover 4 miles at the slower speed, you are actually moving at 30 mi/hr for a longer period of time! Therefore, your average speed is closer to 30 mi/hr than it is to 50 mi/hr. Follow-up: How much farther would you have to drive at 50 mi/hr in order to get back your average speed of 40 mi/hr?

  41. ConcepTest 2.7 Velocity in One Dimension 1) yes 2) no 3) it depends If the average velocity is non-zero over some time interval, does this mean that the instantaneous velocity is never zero during the same interval?

  42. ConcepTest 2.7 Velocity in One Dimension 1) yes 2) no 3) it depends If the average velocity is non-zero over some time interval, does this mean that the instantaneous velocity is never zero during the same interval? No!!! For example, your average velocity for a trip home might be 60 mph, but if you stopped for lunch on the way home, there was an interval when your instantaneous velocity was zero.

  43. vx (m/s) t (sec) Area Interpretation of Displacement The area under a velocity versus time graph (between the curve and the time axis) gives the displacement in a given interval of time. 43

  44. Example :Speedometer readings are obtained and graphed as a car comes to a stop along a straight-line path. How far does the car move between t = 0 and t = 16 seconds? Since there is not a reversal of direction, the area between the curve and the time axis will represent the distance traveled. The rectangular portion has an area of Lw = (20 m/s)(4 s) = 80 m. The triangular portion has an area of ½bh = ½(8 s) (20 m/s) = 80 m. Thus, the total area is 160 m. This is the distance traveled by the car. 44

  45. Acceleration: Rate of Change of Velocity Acceleration is the rate at which velocity changes. • vo initial velocity @ time to • v final velocity @ time t

  46. Average Acceleration • Average Acceleration = change in velocity/time interval aav = Δv/Δt v Δv Slope = Δv/Δt = aav t Δt

  47. (Instantaneous) Acceleration • Instantaneous Acceleration = acceleration @ a given moment in time a= Δv/Δt v Δv slope = a t Δt

  48. Example: The graph shows speedometer readings as a car comes to a stop. What is the acceleration at t = 7.0 s? v The slope of the graph at t = 7.0 sec is 48

  49. Acceleration Acceleration means that the speed of an object is changing, or its direction is, or both.

  50. Constant Acceleration • Constant Acceleration aav = a • In the + Ox direction: • velocity increases a > 0 acceleration • velocity decreases a < 0 deceleration a= Δv/Δt

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