1. Motion in One Dimension

2. Displacement A change in an object’s position:
Δx = xf – xi
This is a vector quantity
Units: m
How is displacement different from distance?

3. Average Velocity and Average Speed
Average Velocity: the displacement of an object divided by the time interval over which that displacement occurred
Vector quantity (sign gives direction)
Units: m/s
Average Speed: the total distance traveled divided by the total time interval to travel that distance
Scalar quantity (always positive)

4. Average Velocity from a Graph
Physics C 1-D Motion
Average Velocity from a Graph A B x Dx Dt t
Bertrand

5. Average Velocity from a Graph
Physics C 1-D Motion
Average Velocity from a Graph A B t x Dx Dt
Bertrand

6. Average Acceleration
Average acceleration of an object is the change in that objects velocity divided by the time interval over which that change occurred
This is how fast the velocity is changing
over time
You can also think about it as:

7. Average Acceleration from a Graph
Physics C 1-D Motion
Average Acceleration from a Graph A B v Dx Dt t
Bertrand

8. Instantaneous Velocity and Speed
Physics C 1-D Motion
Instantaneous Velocity and Speed A t x
Average Velocity from a graph B
Remember that the average velocity between the time at A and the time at B is the slope of the connecting line.
Bertrand

9. Instantaneous Velocity and Speed
Physics C 1-D Motion
Instantaneous Velocity and Speed A t x B
What happens if A and B become closer to each other?
Bertrand

10. Instantaneous Velocity and Speed
Physics C 1-D Motion
Instantaneous Velocity and Speed t x A B
What happens if A and B become closer to each other?
Bertrand

11. Instantaneous Velocity and Speed
Physics C 1-D Motion
Instantaneous Velocity and Speed t x B A
What happens if A and B become closer to each other?
Bertrand

12. Instantaneous Velocity and Speed
Physics C 1-D Motion
Instantaneous Velocity and Speed t x B A
What happens if A and B become closer to each other?
Bertrand

13. Instantaneous Velocity and Speed
Physics C 1-D Motion
Instantaneous Velocity and Speed t x
A and B are effectively the same point. The time difference is effectively zero. B A
The line “connecting” A and B is a tangent line to the curve. The velocity at that instant of time is represented by the slope of this tangent line.
Bertrand

14. Remember that a limit is used to define
a derivative by:

15. Instantaneous Velocity and Speed
Instantaneous velocity is the limiting value of Δx/Δt as Δt approaches zero, or the derivative of x with respect to t:
The instantaneous speed of an object is the magnitude of its velocity

16. Average and Instantaneous Acceleration
Physics C 1-D Motion
Average and Instantaneous Acceleration t v
Instantaneous acceleration is represented by the slope of a tangent to the curve on a v/t graph. A
Average acceleration is represented by the slope of a line connecting two points on a v/t graph. C B
Bertrand

17. Instantaneous Acceleration
Physics C 1-D Motion
Instantaneous Acceleration
Instantaneous acceleration is the limiting value of Δv/Δt as Δt approaches zero, or the derivative of v with respect to t:
Acceleration can also be referred to as the second derivative of position with respect to time.
Just don’t let the new notation scare you; think of the d as a baby Δ, indicating a very tiny change!
Bertrand

18. Now YOU try it!
Try these phun problems using the calculus

22. Derivatives: the graphical representation
Look closely at each of these graphs (all of which are for the same motion)
x vs t, v vs t, and a vs t
Remember that derivatives have to do with slope (over an increasingly small interval)
Analyze each time interval and see if you can understand how these graphs are connected

23. Now YOU try it!
Using your knowledge of how position, velocity, and acceleration are connected, analyze and plot the following graphs

24. Draw representative graphs for a particle which is stationary.
Physics C 1-D Motion
Draw representative graphs for a particle which is stationary. x t
Position vs
time v t
Velocity vs
time a t
Acceleration vs
time
Bertrand

25. Physics C 1-D Motion
Draw representative graphs for a particle which has constant non-zero velocity. x t
Position vs
time v t
Velocity vs
time a t
Acceleration vs
time
Bertrand

26. Physics C 1-D Motion
Draw representative graphs for a particle which has constant non-zero acceleration. x t
Position vs
time v t
Velocity vs
time a t
Acceleration vs
time
Bertrand

27. Given this graph of position vs time, plot the corresponding v vs t and a vs t graphs

28. Given this graph of velocity vs time, plot the corresponding x vs t and a vs t graphs