1. Physics Intro & Kinematics
Quantities
Units
Vectors
Displacement
Velocity
Acceleration
Kinematics
Graphing Motion in 1-D

2. Some Physics Quantities
Vector - quantity with both magnitude (size) and direction
Scalar - quantity with magnitude only
Vectors:
Displacement
Velocity
Acceleration
Momentum
Force
Scalars:
Distance
Speed
Time
Mass
Energy

3. Mass vs. Weight Mass Scalar (no direction)
Measures the amount of matter in an object
Weight
Vector (points toward center of Earth)
Force of gravity on an object
On the moon, your mass would be the same, but the magnitude of your weight would be less.

4. Vectors Vectors are represented with arrows
The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector).
The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle.
5 m/s
42°

5. One of the numbers below does not fit in the group:
Vectors vs. Scalars
One of the numbers below does not fit in the group:
35 ft
161 mph
70° F
200-m, 30° East of North
people

6. Vectors vs. Scalars The answer is: 200-m, 30° East of North
Why is it different?
Numbers w/ magnitude only are called SCALARS.
Numbers w/ magnitude and direction are called VECTORS.

7. Vector Example
Particle travels from A to B along the path shown by the dotted red line
This is the distance traveled and is a scalar
The displacement is the solid line from A to B
displacement is independent of path taken between two points
displacement is a vector
Notice the arrow indicating direction

8. Other Examples of Vectors
Displacement (of 3.5 km at 20o North of East)
Velocity (of 50 km/h due North)
Acceleration (of 9.81 m/s2 downward)
Force (of 10 Newtons in the +x direction)

9. Notation Vectors are written as arrows.
length describes magnitude
direction indicates the direction of vector…
Vectors are written in bold text in your book
Conventions for written notation shown below…

10. Case 1: Collinear Vectors
Adding Vectors
Case 1: Collinear Vectors

11. What is the ground speed of an airplane flying with an air speed of 100 mph into a headwind of 100 mph?

12. Adding Collinear Vectors
When vectors are parallel: just add magnitudes and keep the direction.
Ex:
50 mph east + 40 mph east =
90 mph east

13. Adding Collinear Vectors
When vectors are antiparallel:
just subtract smaller magnitude from larger - use the direction of the larger.
Ex: 50 mph east + 40 mph west = 10 mph east

14. Adding Perpendicular Vectors
When vectors are perpendicular:
sketch the vectors in a HEAD TO TAIL orientation
use right triangle trig to solve for the resultant and direction.
Ex: 50 mph east + 40 mph south = ??

15. An Airplane flies north with an airspeed of 650 mph
An Airplane flies north with an airspeed of 650 mph. If the wind is blowing east at 50 mph, what is the speed of the plane as measured from the ground?

16. Adding Perpendicular Vectors
50 mph R
650 mph θ

17. Examples
Ex1: Find the sum of the forces of 30 lb south and 60 lb east.
Ex2: What is the ground speed of a speed boat crossing a river of 5mph current if the boat can move 20mph in still water?

18. An airplane flies north with an airspeed of 575 mph. 1
An airplane flies north with an airspeed of 575 mph If the wind is blowing 30° north of east at 50 mph, what is the speed of the plane as measured from the ground? What if the wind blew south of west?

19. Adding Skew Vectors
When vectors are not collinear and not perpendicular, we must resort to drawing a scale diagram.
Choose a scale and a indicate a compass
Draw the vectors Head to Tail
Draw the resultant
Measure the resultant and the angle!
Ex: 50 mph east + 40 mph south = ??

20. Adding Skew Vectors R θ
Measure R with a ruler and measure θ with a protractor.

21. Vector Components Vectors are described using their components.
Components of a vector are 2 perpendicular vectors that would add together to yield the original vector.
Components are
notated using
subscripts. F Fy Fx

22. Adding Vectors by Components

23. Adding Vectors by Components
Transform vectors so they are head-to-tail.

24. Adding Vectors by ComponentsBx By A B Ay Ax
Draw components of each vector...

25. Adding Vectors by ComponentsAy Ax Bx
Add components as collinear vectors!

26. Adding Vectors by ComponentsAy Ry Ax Bx Rx
Draw resultants in each direction...

27. Adding Vectors by ComponentsRy q Rx
Combine components of answer using the head to tail method...

28. Adding Vectors Graphically
When you have many vectors, just keep repeating the process until all are included
The resultant is still drawn from the origin of the first vector to the end of the last vector

29. Adding Vectors Graphically, final
Example: A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement? C B A B D D A C R
R is ~2.4 km, 13.5° W of N
or 103.5º from +ve x-axis.

30. Components of a Vector
A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement? Use the component method of vector addition.
Ax = 0 km
Bx = (2 km) cos 45º = 1.4 km
Cx = -4 km
Dx = (3km) cos 45º = 2.1 km
X-components
Y-components
Ay = 3 km
By = (2 km) sin 45º = 1.4 km
Cy = 0 km
Dy = (3km) sin 315º = -2.1 km N y A B C By Bx x W E Dx Dy D S

31. Components of a Vector Ry Rx
Rx = Ax Bx Cx Dx
= 0 km km km km = km
Ry = Ay By Cy + Dy
= 3.0 km km + 0 km km = km
Magnitude: N y R Ry
Direction: x W E Rx
Stop. Think. Is this reasonable? NO! Off by 180º.
Answer: -78º + 180° = 102° S

32. Adding Vectors by Components

33. Components of a Vector
The x-component of a vector is the projection along the x-axis:
The y-component of a vector is the projection along the y-axis:

34. Adding Vectors by Components
Use the Pythagorean Theorem and Right Triangle Trig to solve for R and θ…

35. Units are not the same as quantities!
Quantity Unit (symbol)
Displacement & Distance meter (m)
Time second (s)
Velocity & Speed (m/s)
Acceleration (m/s2)
Mass kilogram (kg)
Momentum (kg · m/s)
Force . . .Newton (N)
Energy Joule (J)

36. SI Prefixes
Little Guys
Big Guys

37. Kinematics definitions
Kinematics – branch of physics; study of motion
Position (x) – where you are located
Distance (d ) – how far you have traveled, regardless of direction
Displacement (x) – where you are in relation to where you started

38. Distance vs. Displacement
You drive the path, and your odometer goes up by 8 miles (your distance).
Your displacement is the shorter directed distance from start to stop (green arrow).
What if you drove in a circle?
start
stop

39. Speed, Velocity, & Acceleration
Speed (v) – how fast you go
Velocity (v) – how fast and which way; the rate at which position changes
Average speed ( v ) – distance / time
Acceleration (a) – how fast you speed up, slow down, or change direction; the rate at which velocity changes

40. Speed vs. Velocity
Speed is a scalar (how fast something is moving regardless of its direction). Ex: v = 20 mph
Speed is the magnitude of velocity.
Velocity is a combination of speed and direction. Ex: v = 20 mph at 15 south of west
The symbol for speed is v.
The symbol for velocity is type written in bold: v or hand written with an arrow: v

41. Speed vs. Velocity
During your 8 mi. trip, which took 15 min., your speedometer displays your instantaneous speed, which varies throughout the trip.
Your average speed is 32 mi/hr.
Your average velocity is 32 mi/hr in a SE direction.
At any point in time, your velocity vector points tangent to your path.
The faster you go, the longer your velocity vector.

42. Acceleration
Acceleration – how fast you speed up, slow down, or change direction; it’s the rate at which velocity changes. Two examples:
t (s)
v (mph) 55 1 57 2 59 3 61
t (s)
v (m/s) 34 1 31 2 28 3 25
a = -3
m / s s
= -3 m / s 2
a = +2 mph / s

43. Velocity & Acceleration Sign Chart+ -
Moving forward; Speeding up
Moving backward; Slowing down
Moving forward; Slowing down
Moving backward; Speeding up

44. Acceleration due to Gravity
Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance).
This acceleration vector is the same on the way up, at the top, and on the way down!
a = -g = -9.8 m/s2
9.8 m/s2
Interpretation: Velocity decreases by 9.8 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.

45. Kinematics Formula Summary
For 1-D motion with constant acceleration:
vf = v0 + a t
vavg = (v0 + vf ) / 2
x = v0 t + ½ a t 2
vf2 – v02 = 2 a x
(derivations to follow)

46. Kinematics Derivations
a = v / t (by definition)
a = (vf – v0) / t
 vf = v0 + a t
vavg = (v0 + vf ) / 2 will be proven when we do graphing.
x = v t = ½ (v0 + vf) t = ½ (v0 + v0 + a t) t  x = v0 t + a t 2
(cont.)

47. Kinematics Derivations (cont.)
vf = v0 + a t  t = (vf – v0) / a x = v0 t a t 2  x = v0 [(vf – v0) / a] + a [(vf – v0) / a]  vf2 – v02 = 2 a x
Note that the top equation is solved for t and that expression for t is substituted twice (in red) into the x equation. You should work out the algebra to prove the final result on the last line.

48. Sample Problems
You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration?
A brick is dropped from 100 m up. Find its impact velocity and air time.
An arrow is shot straight up from a pit 12 m below ground at 38 m/s.
Find its max height above ground.
At what times is it at ground level?

49. Multi-step Problems
How fast should you throw a kumquat straight down from 40 m up so that its impact speed would be the same as a mango’s dropped from 60 m?
A dune buggy accelerates uniformly at 1.5 m/s2 from rest to 22 m/s. Then the brakes are applied and it stops 2.5 s later. Find the total distance traveled.
Answer:
19.8 m/s
Answer: m

50. x t A B C
Graphing !
1 – D Motion
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

51. Graphing w/ Accelerationx
Graphing w/ Acceleration C B t A 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

52. 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

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

54. 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.

55. Times when you are at “home”
Special Points t x Q R P S
Inflection Pt.
P, R
Change of concavity
Peak or Valley Q
Turning point
Time Axis Intercept
P, S
Times when you are at “home”

56. Curve Summary t x B C A D

57. All 3 Graphs t x v t a t

58. Graphing Tips t v t x Line up the graphs vertically.
Draw vertical dashed lines at special points except intercepts.
Map the slopes of the position graph onto the velocity graph.
A red peak or valley means a blue time intercept.

59. Graphing Tips
The same rules apply in making an acceleration graph from a velocity graph. Just graph the slopes! Note: a positive constant slope in blue means a positive constant green segment. The steeper the blue slope, the farther the green segment is from the time axis. v t a t

60. 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 green segments would be connected. In our class, however, we’ll mainly deal with constant acceleration. v t a t

61. Area under a velocity graph
“forward area”
“backward area”
Area above the time axis = forward (positive) displacement.
Area below the time axis = backward (negative) displacement.
Net area (above - below) = net displacement.
Total area (above + below) = total distance traveled.

62. Area “forward area” “backward area” t v t
The areas above and below 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 too. t x

63. v (m/s)
Area units 12
t (s)
Imagine approximating the area under the curve with very thin rectangles.
Each has area of height  width.
The height is in m/s; width is in seconds.
Therefore, area is in meters!
12 m/s
0.5 s
The rectangles under the time axis have negative heights, corresponding to negative displacement.

64. Graphs of a ball thrown straight upx
The ball is thrown from the ground, and it lands on a ledge.
The position graph is parabolic.
The ball peaks at the parabola’s vertex.
The v graph has a slope of -9.8 m/s2.
Map out the slopes!
There is more “positive area” than negative on the v graph. t v t a t

65. Graph Practice Try making all three graphs for the following scenario:
1. Schmedrick 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, Schmedrick 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.

66. Kinematics Practice
A catcher catches a 90 mph fast ball. His glove compresses 4.5 cm. How long does it take to come to a complete stop? Be mindful of your units!
2.24 ms
Answer

67. Uniform Acceleration
x = 1
x = 3
x = 5
x = 7
t :
x :
( arbitrary units )
When object starts from rest and undergoes constant acceleration:
Position is proportional to the square of time.
Position changes result in the sequence of odd numbers.
Falling bodies exhibit this type of motion (since g is constant).

68. Spreadsheet Problem
We’re analyzing position as a function of time, initial velocity, and constant acceleration.
x, x, and the ratio depend on t, v0, and a.
x is how much position changes each second.
The ratio (1, 3, 5, 7) is the ratio of the x’s.
Make a spreadsheet like this and determine what must be true about v0 and/or a in order to get this ratio of odd numbers.
Explain your answer mathematically.

69. Let’s use the kinematics equations to answer these:
Relationships
Let’s use the kinematics equations to answer these:
1. A mango is dropped from a height h.
a. If dropped from a height of 2 h, would the impact speed double?
Would the air time double when dropped from a height of 2 h ?
A mango is thrown down at a speed v.
If thrown down at 2 v from the same height, would the impact speed double?
Would the air time double in this case?

70. Relationships (cont.)
A rubber chicken is launched straight up at speed v from ground level. Find each of the following if the launch speed is tripled (in terms of any constants and v).
max height
hang time
impact speed
9 v2 / 2 g
6 v / g
3 v
Answers