1. Motion in One Dimension
Chapter 2
Motion in One Dimension

2. Kinematics
Describes motion while ignoring the agents that caused the motion
For now, will consider motion in one dimension
Along a straight line
Will use the particle model
A particle is a point-like object, has mass but infinitesimal size

3. Position Defined in terms of a frame of reference
One dimensional, so generally the x- or y-axis
The object’s position is its location with respect to the frame of reference

4. Position-Time Graph
The position-time graph shows the motion of the particle (car)
The smooth curve is a guess as to what happened between the data points

5. Active Figure
AF_0201 position versus time.swf

6. Displacement
Defined as the change in position during some time interval
Represented as x
SI units are meters (m) x can be positive or negative
Different than distance – the length of a path followed by a particle

7. Vectors and Scalars
Vector quantities need both magnitude (size or numerical value) and direction to completely describe them
Will use + and – signs to indicate vector directions
Scalar quantities are completely described by magnitude only

8. Average Velocity
The average velocity is rate at which the displacement occurs
The dimensions are length / time [L/T]
The SI units are m/s
Is also the slope of the line in the position – time graph

9. Average Velocity, cont Gives no details about the motion
Gives the result of the motion
It can be positive or negative
It depends on the sign of the displacement
It can be interpreted graphically
It will be the slope of the position-time graph

10. Displacement from a Velocity - Time Graph
The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve

11. Instantaneous Velocity
The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero
The instantaneous velocity indicates what is happening at every point of time

12. Instantaneous Velocity, graph
The instantaneous velocity is the slope of the line tangent to the x vs. t curve
This would be the green line
The blue lines show that as t gets smaller, they approach the green line

13. Active Figure
AF_0202 velocity and slope.swf

14. Instantaneous Velocity, equations
The general equation for instantaneous velocity is
When “velocity” is used in the text, it will mean the instantaneous velocity

15. Instantaneous Velocity, Signs
At point A, vx is positive
The slope is positive
At point B, vx is zero
The slope is zero
At point C, vx is negative
The slope is negative

16. Instantaneous Speed
The instantaneous speed is the magnitude of the instantaneous velocity vector
Speed can never be negative

17. Constant Velocity If the velocity of a particle is constant
Its instantaneous velocity at any instant is the same as the average velocity over a given time period
vx = v x avg = Dx / Dt
Also, xf = xi + vx t
These equations can be applied to particles or objects that can be modeled as a particle moving under constant velocity

18. Average Acceleration
Acceleration is the rate of change of the velocity
It is a measure of how rapidly the velocity is changing
Dimensions are L/T2
SI units are m/s2

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

20. Instantaneous Acceleration – Graph
The slope of the velocity vs. time graph is the acceleration
Positive values correspond to where velocity in the positive x direction is increasing
The acceleration is negative when the velocity is in the positive x direction and is decreasing

21. Negative Acceleration
Negative acceleration does not necessarily mean that an object is slowing down
If the velocity is negative and the acceleration is negative, the object is speeding up
Deceleration is commonly used to indicate an object is slowing down, but will not be used in this text

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

23. Acceleration and Velocity, 2
The car is moving with constant positive velocity (shown by red arrows maintaining the same size)
Acceleration equals zero

24. Acceleration and Velocity, 3
Velocity and acceleration are in the same direction
Acceleration is uniform (blue arrows maintain the same length)
Velocity is increasing (red arrows are getting longer)
This shows positive acceleration and positive velocity

25. Acceleration and Velocity, 4
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

26. Active Figure AF_0208 match the motion.swf
AF_0212 acceleration and slope.swf
AF_0211 motion graphs for an accelerating car.swf

27. Kinematic Equations – Summary

28. 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
The equations are useful when you can model a situation as a particle under constant acceleration

29. 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

30. Kinematic Equations, specific
For constant acceleration,
The average velocity can be expressed as the arithmetic mean of the initial and final velocities
Only valid when acceleration is constant

31. Kinematic Equations, specific
For constant acceleration,
This gives you the position of the particle in terms of time and velocities
Doesn’t give you the acceleration

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

33. Kinematic Equations, specific
For constant a,
Gives final velocity in terms of acceleration and displacement
Does not give any information about the time

34. 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

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

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

37. Problem-Solving Strategy –Constant Acceleration
Conceptualize
Establish a mental representation
Categorize
Confirm there is a particle or an object that can be modeled as a particle
Confirm it is moving with a constant acceleration

38. Problem-Solving Strategy –Constant Acceleration, cont
Analyze
Set up the mathematical representation
Choose the instant to call the initial time and another to be the final time
These are defined by the parts of the problem of interest, not necessarily the actual beginning and ending of the motion
Choose the appropriate kinematic equation(s)

39. Problem-Solving Strategy –Constant Acceleration, Final
Finalize
Check to see if the answers are consistent with the mental and pictorial representations
Check to see if your results are realistic

40. 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

41. 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/s2
g decreases with increasing altitude
g varies with latitude
9.80 m/s2 is the average at the earth’s surface

42. Acceleration of Free Fall, cont.
We will neglect air resistance
Free fall motion is constantly accelerated motion in one dimension
Let upward be positive
Use the kinematic equations with ay = g = m/s2
The negative sign indicates the direction of the acceleration is downward

43. Free Fall Example
Initial velocity at A is upward (+) and acceleration is g (-9.8 m/s2)
At B, the velocity is 0 and the acceleration is g (-9.8 m/s2)
At C, the velocity has the same magnitude as at A, but is in the opposite direction
The displacement is –50.0 m (it ends up 50.0 m below its starting point)

44. Free Fall Terminal Velocity
AF_0518 terminal speed.swf