1. A Mathematical Model of Motion
PHYSICS

2. 1 Graphing Motion in One Dimension
Interpret graphs of position versus time for a moving object to determine the velocity of the object
Describe in words the information presented in graphs and draw graphs from descriptions of motion
Write equations that describe the position of an object moving at constant velocity

3. Parts of a Graph
X-axis
Y-axis
All axes must be labeled with appropriate units, and values.

4. Position vs. Time The x-axis is always “time”
The y-axis is always “position”
The slope of the line indicates the velocity of the object.
Slope = (y2-y1)/(x2-x1)
d1-d0/t1-t0
Δd/Δt

5. Uniform Motion
Uniform motion is defined as equal displacements occurring during successive equal time periods (sometimes called constant velocity)
Straight lines on position-time graphs mean uniform motion.

6. Given below is a diagram of a ball rolling along a table
Given below is a diagram of a ball rolling along a table. Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds.
Notice that the ball covers an equal distance between flashes. Let's assume this distance equals 20 cm and display the ball's behavior on a graph plotting its x-position versus time.

7. The slope of this line would equal 20 cm divided by 0
The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This represents the ball's average velocity as it moves across the table. Since the ball is moving in a positive direction its velocity is positive. That is, the ball's velocity is a vector quantity possessing both magnitude (200 cm/sec) and direction (positive).

8. Steepness of slope on Position-Time graph
Slope is related to velocity
Steep slope = higher velocity
Shallow slope = less velocity

9. Different Position. Vs. Time graphs
Accelerated Motion
Uniform Motion
Constant positive velocity
(zero acceleration)
Increasing positive velocity
(positive acceleration)
Constant negative velocity
(zero acceleration)
Decreasing negative velocity
(positive acceleration)

10. Different Position. Vs. Time
Changing slope means changing velocity!!!!!!
Increasing negative slope = ??
Decreasing negative slope = ??

11. X t B A C A … Starts at home (origin) and goes forward slowly
B … Not moving (position remains constant as time progresses)
C … Turns around and goes in the other direction quickly, passing up home

12. During which intervals was he traveling in a positive direction?
During which intervals was he traveling in a negative direction?
During which interval was he resting in a negative location?
During which interval was he resting in a positive location?
During which two intervals did he travel at the same speed?
A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec

13. Graphing w/ Accelerationx
Graphing w/ Acceleration t A B C D
A … Start from rest south of home; increase speed gradually
B … Pass home; gradually slow to a stop (still moving north)
C … Turn around; gradually speed back up again heading south
D … Continue heading south; gradually slow to a stop near the starting point

14. Tangent Lines t On a position vs. time graph: SLOPE VELOCITY Positivex
Tangent Lines t
On a position vs. time graph:
SLOPE
VELOCITY
Positive
Negative
Zero
SLOPE
SPEED
Steep
Fast
Gentle
Slow
Flat
Zero

15. Increasing & Decreasingt x
Increasing
Decreasing
On a position vs. time graph:
Increasing means moving forward (positive direction).
Decreasing means moving backwards (negative direction).

16. Concavity On a position vs. time graph:x
Concavity
On a position vs. time graph:
Concave up means positive acceleration.
Concave down means negative acceleration.

17. Special Points t Inflection Pt. P, Rx Q R P S
Inflection Pt.
P, R
Change of concavity, change of acceleration
Peak or Valley Q
Turning point, change of positive velocity to negative
Time Axis Intercept
P, S
Times when you are at “home”, or at origin

18. 2 Graphing Velocity in One Dimension
Determine, from a graph of velocity versus time, the velocity of an object at a specific time
Interpret a v-t graph to find the time at which an object has a specific velocity
Calculate the displacement of an object from the area under a v-t graph

19. Velocity vs. Time X-axis is the “time” Y-axis is the “velocity”
Slope of the line = the acceleration

20. Different Velocity-time graphs

21. Different Velocity-time graphs

22. Velocity vs. Time Horizontal lines = constant velocity
Sloped line = changing velocity
Steeper = greater change in velocity per second
Negative slope = deceleration

23. Acceleration vs. Time Time is on the x-axis
Acceleration is on the y-axis
Shows how acceleration changes over a period of time.
Often a horizontal line.

24. All 3 Graphs t x v t a t

25. Real life
Note how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the orange segments would be connected. In our class, however, we’ll only deal with constant acceleration. v t a t

26. Constant Rightward Velocity

27. Constant Leftward Velocity

28. Constant Rightward Acceleration

29. Constant Leftward Acceleration

30. Leftward Velocity with Rightward Acceleration

31. Graph Practice Try making all three graphs for the following scenario:
1. Newberry starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed.
2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose.
3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail.
4. Fleeing the scene of the crime, Newberry takes off again in the same direction, speeding up quickly.
5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.

32. Area Underneath v-t Graph
If you calculate the area underneath a v-t graph, you would multiply height X width.
Because height is actually velocity and width is actually time, area underneath the graph is equal to
Velocity X time or
V·t

33. Remember that Velocity = Δd Rearranging, we get Δd = velocity X Δt
So….the area underneath a velocity-time graph is equal to the displacement during that time period.

34. Area “positive area” “negative area” t v t
Note that, here, the areas are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this as well. t x

35. Velocity vs. Time
The area under a velocity time graph indicates the displacement during that time period.
Remember that the slope of the line indicates the acceleration.
The smaller the time units the more “instantaneous” is the acceleration at that particular time.
If velocity is not uniform, or is changing, the acceleration will be changing, and cannot be determined “for an instant”, so you can only find average acceleration

36. 3 Acceleration
Determine from the curves on a velocity-time graph both the constant and instantaneous acceleration
Determine the sign of acceleration using a v-t graph and a motion diagram
Calculate the velocity and the displacement of an object undergoing constant acceleration

37. Acceleration
Like speed or velocity, acceleration is a rate of change, defined as the rate of change of velocity
Average Acceleration = change in velocity
Elapsed time
Units of acceleration?

38. Rearrangement of the equation

39. Finally…
This equation is to be used to find (final) velocity of an accelerating object. You can use it if there is or is not a beginning velocity

40. Displacement under Constant Acceleration
Remember that displacement under constant velocity was
Δd = vt or d1 = d0 + vt
With acceleration, there is no one single instantaneous v to use, but we could use an average velocity of an accelerating object.

41. Average velocity of an accelerating object would simply be ½ of sum of beginning and ending velocities
Average velocity of an accelerating object
V = ½ (v0 + v1)

42. So…….
Key equation

43. Some other equations 2
This equation is to be used to find final position when there is an initial velocity, but velocity at time t1 is not known.

44. If no time is known, use this to find final position….2 2

45. Finding final velocity if no time is known…2 2

46. The equations of importance

47. 2 2 2 2 2

48. Practical Application Velocity/Position/Time equations
Calculation of arrival times/schedules of aircraft/trains (including vectors)
GPS technology (arrival time of signal/distance to satellite)
Military targeting/delivery
Calculation of Mass movement (glaciers/faults)
Ultrasound (speed of sound) (babies/concrete/metals) Sonar (Sound Navigation and Ranging)
Auto accident reconstruction
Explosives (rate of burn/expansion rates/timing with det. cord)

49. 4 Free Fall Recognize the meaning of the acceleration due to gravity
Define the magnitude of the acceleration due to gravity as a positive quantity and determine the sign of the acceleration relative to the chosen coordinate system
Use the motion equations to solve problems involving freely falling objects

50. Freefall
Defined as the motion of an object if the only force acting on it is gravity.
No friction, no air resistance, no drag

51. Acceleration Due to Gravity
Galileo Galilei recognized about 400 years ago that, to understand the motion of falling objects, the effects of air or water would have to be ignored.
As a result, we will investigate falling, but only as a result of one force, gravity.
Galileo Galilei

52. Galileo’s Ramps
Because gravity causes objects to move very fast, and because the time-keepers available to Galileo were limited, Galileo used ramps with moveable bells to “slow down” falling objects for accurate timing.

53. Galileo’s Ramps To keep “accurate” time, Galileo used a water clock.
For the measurement of time, he employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which he collected in a small glass during the time of each descent... the water thus collected was weighed, after each descent, on a very accurate balance; the difference and ratios of these weights gave us the differences and ratios of the times...

55. Displacements of Falling Objects
Looking at his results, Galileo realized that a falling ( or rolling downhill) object would have displacements that increased as a function of the square of the time, or t2
Another way to look at it, the velocity of an object increased as a function of the square of time, multiplied by some constant.

56. Galileo also found that all objects, no matter what slope of ramp he rolled them down, and as long as the ramps were all the same height, would reach the bottom with the same velocity.

57. Galileo’s Finding
Galileo found that, neglecting friction, all freely falling objects have the same acceleration.

58. Acceleration Due to Gravity
Galileo calculated that all freely falling objects accelerate at a rate of
9.8 m/s2
This value, as an acceleration, is known as g

59. Acceleration Due to Gravity
Because this value is an acceleration value, it can be used to calculate displacements or velocities using the acceleration equations learned earlier. Just substitute g for the a

60. Example problem A brick is dropped from a high building.
What is it’s velocity after 4.0 sec.?
How far does the brick fall during this time?