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Kinematics in One Dimension: Understanding Displacement, Velocity, and Acceleration

Learn about position, displacement, distance, vectors, scalars, speed, velocity, and their significance in one-dimensional kinematics. Practice problems included.

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Kinematics in One Dimension: Understanding Displacement, Velocity, and Acceleration

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  1. Chapter 2: Kinematics in one Dimension Displacement Velocity Acceleration

  2. Chapter 2: Kinematics in one Dimension Coordinate Axis • In Physics we draw a set ofcoordinate axisto represent a frame of reference. • In one dimensional axis coordinate, the position of an object is given by its x or y. y x -x o -y

  3. Position on a line Reference point (origin) position Distance Direction • The position of Charlotte in reference to Fort Mill ( Fort Mill is the origin) • Symbol for position: x • SI units: meters, m

  4. Displacement on a line • Change of position is called Displacement: xf xi Displacement is a vector quantity It has magnitude and direction

  5. Displacement • Defined as the change in position during some time interval • Represented as x • SI units are meters (m) x can be positive or negative • Different than distance – the length of a path followed by a particle. • Displacement has both a magnitude and a direction so it is a vector.

  6. Example • Mary walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the displacement and distance Mary travelled.

  7. Vectors and Scalars • Vector quantities need both a magnitude (size or numerical value) and direction to completely describe them • Will use + and - signs to indicate vector directions • Scalars quantities are completely described by magnitude only

  8. Average Speed • Average speed =distance traveled/ time elapsed • Example: if a car travels 300 kilometer (km) in 2 hours (h), its average speed is 150km/h. • Not to confuse with average velocity. • Average speed is a scalar

  9. Average Velocity • The average velocity is rate at which the displacement occurs • The SI units are m/s • Is also the slope of the line in the position – time graph

  10. Average Velocity, cont • Gives no details about the motion • Gives the result of the motion • It can be positive or negative • It depends on the sign of the displacement • It can be interpreted graphically • It will be the slope of the position-time graph

  11. Average Speed and Average Velocity Speed is how far an object travels in a given time interval: Velocity includes directional information:

  12. Not to Confuse • Speed is a number : a scalar • Velocity is a vector : with a magnitude and a direction

  13. Average velocity from a graph of x(t) x2 x1 t1 t2 v(t) = slope of x(t) Position (x) Time (t)

  14. Average Speed Example 2-2: Distance a cyclist travels. How far can a cyclist travel in 2.5 h along a straight road if her average speed is 18 km/h?

  15. Example 1 • While on Vacation Lisa traveled a total distance of 440 miles her trip took 8 h , what was her average speed?

  16. Example 1 solution

  17. Example 2 • Mary walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed and the average velocity.

  18. Mary walked a distanceof 12 meters in 24 seconds; thus, her average speed was 0.50 m/s. • However, since her displacement is 0 meters, her average velocity is 0 m/s. • Remember that the displacement refers to the change in position and the velocity is based upon this position change. In this case of the Mary’s motion, there is a position change of 0 meters and thus an average velocity of 0 m/s.

  19. Problem 6: Average Speed • 6. (II) You are driving home from school steadily at 95 km/h for 130 km. It then begins to rain and you slow to 65 km/h. You arrive home after driving 3 hours and 20 minutes. (a) How far is your hometown from school? (b) What was your average speed?

  20. Instantaneous Velocity The instantaneous velocity is the average velocity in the limit as the time interval becomes infinitesimally short. Ideally, a speedometer would measure instantaneous velocity; in fact, it measures average velocity, but over a very short time interval.

  21. Instantaneous velocity from a graph of x(t) Sign + slope - slope x P1 P2 Position (x) v(t) = slope of x(t) t t Time (t) +direction -direction

  22. Instantaneous Velocity The instantaneous speed always equals the magnitude of the instantaneous velocity; But it only equals the average velocity if the velocity is constant (graph (a)). (a) (b)

  23. Instantaneous Velocity Example 2-3: Given x as a function of t. A jet engine moves along an experimental track (which we call the x axis) as shown. We will treat the engine as if it were a particle. Its position as a function of time is given by the equation x = At 2 + B, where A = 2.10 m/s2 and B = 2.80 m. (a) Determine the displacement of the engine during the time interval from t1 = 3.00 s to t2 = 5.00 s. (b) Determine the average velocity during this time interval. (c) Determine the magnitude of the instantaneous velocity at t = 5.00 s.

  24. Instantaneous Velocity: Problem 16

  25. Instantaneous Velocity:Problem 8

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