2. Moving Experiences: A Graphical Approach to Position, Velocity and Acceleration

3. An object starts at position x i and travels to position x f in a time interval  t xixi xfxf titi tftf We represent this motion with a position-time graph, with position on the vertical axis and time on the horizontal.

4. Define the object’s average velocity during the interval  t: It should be clear that this average velocity is also the slope of the object’s position-time graph. xixi xfxf titi tftf

5. Note that average velocity over an interval is signed Positive velocity is going forwards. xixi xfxf titi tftf

6. And the sign of velocity is relative to the position coordinate system Negative velocity is going backwards! xfxf xixi titi tftf

7. Displacement is change in position Positive v sets x f > x i. xixi xfxf titi tftf

8. Displacement can be negative Negative v sets x f < x i xfxf xixi titi tftf

9. Displacement varies in sign, but distance traveled does not xfxf xixi titi tftf Displacement: Distance traveled: Distance traveled: Same same

10. This distinction is especially important for a round trip xixi xfxf titi tftf Distance traveled is 2|x f - x i | Displacement = 0

11. Speed is the magnitude of velocity, which cannot be negative xixi xfxf titi tftf Round trip speed is 2|x f - x i | (t f – t i ) Displacement = 0

12. Now its time to accelerate

13. Suppose velocity changes at the constant rate a, such that:

14. The average velocity during a time interval  t:

15. Substitute this form of the average velocity back into the expression for x f

16. Thus: Three Equations of Motion for constant acceleration during a time interval t

17. For constant acceleration, distance traveled is a quadratic function of time:

18. Another way: Velocity-time graphs vivi vfvf titi tftf How far does an object traveling at constant v i go in time  t? Ans:  x = v i  t

19. Ans:  x = v f  t vivi vfvf titi tftf How far does an object traveling at constant v f go in time  t?

20.  x = v  t vivi vfvf titi tftf Both distances are numerically equal to the area of the rectangle of height v and width  t.

21. Suppose velocity changes by constant acceleration: vivi vfvf titi tftf How far does an object accelerating from v i to v f go in time  t?

22. Ans: x = v i  t + 1/2 (v f - v i )  t vivi vfvf titi tftf How far does an object accelerating from v i to v f go in time  t?

23. But v f - v i = a  t! vivi vfvf titi tftf So:  x = v i  t + 1/2 a  t 2 Note that the velocity graph tells us nothing about the initial x i

24. These relationships between a function, its slope and the area below its graph vivi vfvf titi tftf are the key ties between the Physics of Motion and the Calculus

25. But we can still produce one more equation!

26. One more equation:

27. The 4 Equations of Motion for Constant Acceleration