1. Describing Motion Chapter 3

2. What is a motion diagram?  A Motion diagram is a useful tool to study the relative motion of objects.  From motion diagrams, it is possible to observe an object under:  Constant velocity  Accelerating positively  Accelerating negatively  Or Stationary

3. Motion Diagrams Constant Speed: Negative Acceleration: Positive Acceleration:

4. The Particle Model TTTTo simplify motion diagrams, we can concentrate all the motion through a single point at or near the center of gravity.

5. Constant Speed: Negative Acceleration: Positive Acceleration:

6. The Particle Model Constant Speed: Negative Acceleration: Positive Acceleration:

7. Determining Motion  An object’s motion can be determined if its initial and subsequent positions are identified relative to time. Initial Time = t i Initial Position = d i Initial Velocity= v i Final Time = t f Final Position = d f Final Velocity= v f

8. Average Velocity  The average velocity is the ratio of displacement and time as follows: Where:  d = the displacement vector  t = change in time t i and d i represent the starting position t f and d f represent the final position  Average velocity does not tell you how the velocity varied during the time interval between the points, d i and d f. v = = ddttddtt d f - d i t f - t i t f - t i (1)

9. Graphical Representation of Velocity  A graph of an object moving at constant velocity will consist of a straight line.  The slope of this line will equal the average velocity of the object.

10. Average Acceleration  An object in motion with changing velocity is under acceleration  Acceleration is the rate of change of velocity as follows:  As with average velocity, the average acceleration does not tell you how it varied during the time interval t i to t f. a = = vvttvvtt v f - v i t f - t i t f - t i(2)

11. Graphical Representation of Average Acceleration  A graph of an object moving at constant acceleration will consist of a straight line.  The slope of this line will equal the average acceleration of the object.  The average between the initial and final values for velocity will equal the average. vivi vfvf v avg

12. Finding Final Velocity Under Uniform Acceleration  To find the final velocity when acceleration is uniform, all that is needed is the initial velocity, acceleration and time.  By rearranging 2 to isolate v f, we obtain:  An alternative method for calculating the final velocity is:  t v f = v i + a  t (3)  d v f 2 = v i 2 + 2a  d (4)

13. Average Velocity during Uniform Acceleration  For an object moving at constant acceleration, the average velocity is equal to the average of the initial plus final velocities. v avg = v i + v f 2 (5)

14. Finding Displacement Under Uniform Acceleration  When acceleration is uniform, the displacement depends on the objects acceleration, initial velocity and time.  To find the displacement of an object during uniform acceleration, substitute 1 into 5 for v avg. v avg =  d/  t (1) v avg = v i + v f 2(5)  d = (v i + v f )  t (6) 1212  d/  t = v i + v f 2

15. Finding Displacement Under Uniform Acceleration  An alternative expression for (6) can be obtained by substituting 3 into 6:  d = v i  t + a(  t) 2 (7) 1212  d = (v i + v f )  t (6) 1212  d = (v i + v i + a  t)  t 1212  d = [2v i  t + a(  t) 2 ] 1212 v f = v i + a  t (3)

16. Formulas for Motion of Objects Equations to use when an accelerating object has an initial velocity. Form to use when accelerating object starts from rest (v i = 0).  d = ½ (v i + v f )  t  d = ½ v f  t v f = v i + a  t v f = a  t  d = v i  t + ½ a(  t) 2  d = ½ a(  t) 2  d v f 2 = v i 2 + 2a  d  d v f 2 = 2a  d

17. Formulas for Motion of Objects assuming d is displacement from origin and time starts at 0. Equations to use when an accelerating object has an initial velocity. Form to use when accelerating object starts from rest (v i = 0). d = ½ (v i + v f ) t = v ave t d = ½ v f t v f = v i + at v f = at d = v i t + ½ a(t) 2 d = ½ a(t) 2 vf2 = vi2 + 2advf2 = vi2 + 2advf2 = vi2 + 2advf2 = vi2 + 2ad vf2 = 2advf2 = 2advf2 = 2advf2 = 2ad