1. Chapter 2 Kinematics in one Dimension June 4, 2016

2. Going around in circles … A B s=pathlength Seconds x=displacement x A B t=time interval (A,B) = 10 seconds

3. A B s x DEFINITIONS s=path length or total distance traveled. x=displacement = NET distance traveled. Difference between final and initial position independent of the path. t=time of the trip. Usually measured in seconds.

4. More Definitions average speed = distance traveled (s) / time taken [meters/sec.] average velocity= displacement/time [m/s] Note: the actual SYMBOL used for distance and displacement will usually change with the context of the problem.

5. 10s -40m

6. In the limit - xx tt In the limit of  t  0, velocity is the slope of the tangent to the x-t curve!

7. For v=constant

8. Constant Velocity v xixi xfxf x=0 x f – x i = distance traveled = v t MOTION DOESN’T ALWAYS START AT THE ORIGIN

9. Question The position of a particle as a function of time is given by the equation below. (a) What is the position of the particle at t=2 seconds? (b) What is the velocity of the particle at t= 2 seconds? (c) At what time is the particle’s velocity = 0? (d) Is the particle ever at the origin? When?

10. Conclusion Velocity is not the same at all times. It is changing A changing velocity is called an acceleration.

11. Acceleration in Pictures Constant Velocity Constant Acceleration

12. Average Acceleration Acceleration is the rate of change of the velocity. Dimensions are L/t 2 SI units are m/s²

13. Example – Constant acceleration a=1 m/s 2 Time (s)Velocity (m/s) 00 11 22 33 44 55 66 77 88 99 10 11 12 13 14 15 16 17 18 19 20

14. Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as  t approaches 0

15. Calculus

16. Instantaneous Acceleration -- graph The slope of the velocity vs. time graph is the acceleration The green line represents the instantaneous acceleration The blue line is the average acceleration

17. Acceleration and Velocity, 1 When an object’s velocity and acceleration are in the same direction, the object is speeding up When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

18. Example Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration

19. Some math. Assume constant acceleration

20. More …

21. the chain rule One more calculus trick: the chain rule

22. Kinematic Equations -- summary from the book.

23. Kinematic Equations The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration You may need to use two of the equations to solve one problem Many times there is more than one way to solve a problem

24. Kinematic Equations, specific For constant a,  Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration  Does not give any information about displacement

25. Kinematic Equations, specific For constant acceleration, The average velocity can be expressed as the arithmetic mean of the initial and final velocities

26. Kinematic Equations, specific For constant acceleration, Gives final position in terms of velocity and acceleration Doesn’t tell you about final velocity

27. Graphical Look at Motion – displacement – time curve The slope of the curve is the velocity The curved line indicates the velocity is changing  Therefore, there is an acceleration

28. Graphical Look at Motion – velocity – time curve The slope gives the acceleration The straight line indicates a constant acceleration

29. The zero slope indicates a constant acceleration Graphical Look at Motion – acceleration – time curve

30. Let’s get real and throw something out of a building and watch it fall. This is called free fall.

31. Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity alone. It does not depend upon the initial motion of the object  Dropped – released from rest  Thrown downward  Thrown upward

32. Acceleration of Freely Falling Object The acceleration of an object in free fall is directed downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80 m/s 2  g decreases with increasing altitude  g varies with latitude  9.80 m/s 2 is the average at the Earth’s surface In the English/American system, g=32 ft/sec 2.

33. Acceleration of Free Fall/Rise, cont. We will neglect air resistance Free fall motion is constantly accelerated motion in one dimension Use the kinematic equations to solve problems.

34. Free Fall Example Let’s drop an object from the top of this building by simply releasing it. How long will it take to get to the ground? How fast will it be going when it gets to the ground? What color is the object? Issues – Where is the origin? IS y the same thing as x?? How can that be???