1. Motion in One Dimension (Velocity/Speed vs. Time) Chapter 5.2

2. What is instantaneous speed?

3. What effect does an increase in speed have on displacement? d1d1 d2d2 d3d3 d4d4 d5d5 d6d6

4. Instead of position vs. time, consider velocity or speed vs. time. High acceleration Relatively constant speed = no acceleration

5. What is the significance of the slope of the velocity/speed vs. time curve? Since velocity is on the y-axis and time is on the x-axis, it follows that the slope of the line would be: Therefore, slope must equal acceleration. Time

6. What information does the slope of the velocity vs. time curve provide? A.Positively sloped curve = increasing velocity (Speeding up). B.Negatively sloped curve = decreasing velocity (Slowing down). C.Horizontally sloped curve = constant velocity. Time Positive Acceleration A Time Zero Acceleration C Time Negative Acceleration B

7. Acceleration determined from the slope of the curve. rise run v f – v i t f – t i 8.4m/s-0m/s 1.7s-0.00s m = 4.9 m/s 2 Since m = a: a = 4.9 m/s 2 m = Slope = m = What is the acceleration from t = 0 to t = 1.7 seconds?

8. How can displacement be determined from a v vs. t graph? Measure the area under the curve. d = v*t Where t is the x component v is the y component Time A2A2 A1A1 A 1 = d 1 = ½  v 1 *  t 1 A 2 = d 2 = v 2 *  t 2 d total = d 1 + d 2

9. Measuring displacement from a velocity vs. time graph. A = ½ b x h A = ½ (2.36s)(11.7m/s) A = 13.8 m A = b x h A = (7.37s)(11.7m/s) A = 86.2 m

10. Algebraically deriving the kinematics formulas in your reference table

11. Determining velocity from acceleration Time

12. What is the average velocity? Time This latter formula is not in your reference table!

13. How to determine position, velocity or acceleration without time. dd

14. How to determine displacement, time or initial velocity without the final velocity.

15. 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).

16. Acceleration due to Gravity All falling bodies accelerate at the same rate when the effects of friction due to water, air, etc. can be ignored. Acceleration due to gravity is caused by the influences of Earth’s gravity on objects. The acceleration due to gravity is given the special symbol g. The acceleration of gravity is a constant close to the surface of the earth. g = 9.81 m/s 2

17. Example 1: Calculating Distance A stone is dropped from the top of a tall building. After 3.00 seconds of free-fall, what is the displacement, y of the stone? Data y ? a = g -9.81 m/s 2 vfvf n/a vivi 0 m/s t 3.00 s

18. Example 1: Calculating Distance

19. Example 2: Calculating Final Velocity What will the final velocity of the stone be? Data y -44.1 m a = g -9.81 m/s 2 vfvf ? vivi 0 m/s t 3.00 s

20. Example 2: Calculating Final Velocity

21. Example 3: Determining the Maximum Height How high will the coin go? Data y ? a = g -9.81 m/s 2 vfvf 0 m/s vivi 5.00 m/s t ?

22. Example 3: Determining the Maximum Height

23. Example 4: Determining the Total Time in the Air How long will the coin be in the air? Data y 1.27 m a = g -9.81 m/s 2 vfvf 0 m/s vivi 5.00 m/s t ?

24. Example 4: Determining the Total Time in the Air

25. Key Ideas Instantaneous velocity is equal to the slope of a line tangent to a position vs. time graph. Slope of a velocity vs. time graphs provides an objects acceleration. The area under the curve of a velocity vs. time graph provides the objects displacement. Acceleration due to gravity is the same for all objects when the effects of friction due to wind, water, etc can be ignored.

26. Important equations to know for uniform acceleration. d f = d i + ½ (v i + v f )*t d f = d i + v i t + ½ at 2 v f 2 = v i 2 + 2a*(d f – d i ) v f = v i +at a = Δv/Δt = (v f – v i )/(t f – t i )

27. Determining instantaneous velocity 1997 World Championships - Athens, Greece Maurice Green 0 10 20 30 40 50 60 70 80 90 100 0246810 Time (s) Distance (m) y = 1.13x + 4.08x - 0.05 R 2 = 1.00 2 y = 11.65x - 13.07 R = 1.00 2

28. How do you determine the instantaneous velocity? Instantaneous velocity = slope of line tangent to curve. What is the runners velocity at t = 1.5s?

29. Determining the instantaneous velocity from the slope of the curve. m = rise/run m = 25m – 5 m 3.75s – 1.0s m = 7.3 m/s v = 7.3 m/s @ 1.5s

30. Acceleration determined from the slope of the curve. rise run v f – v i t f – t i 13m/s-7m/s 3.75s-0.75s m = 2.0 m/s 2 Since m = a: a = 2.0 m/s 2 m = Slope = m = What is the acceleration at t = 2 seconds?

31. Displacement when acceleration is constant. Displacement = area under the curve. Δd = v i t + ½ (v f – v i )*t Simplifying: Δd = ½ (v f + v i )*t If the initial position, d i, is not 0, then: d f = d i + ½ (v f + v i )*t By substituting v f = v i + at d f = d i + ½ (v i + at + v i )*t Simplifying: d f = d i + v i t + ½ at 2 d = v i t d = ½ (v f -v i )t vfvf vivi t