1. Vectors and Linear Motion

2. Linear Motion Motion in one dimension Distance (x) Time (t) Speed (v)
This dimension could be horizontal or it could be vertical
An object thrown up in the air undergoes linear motion
So does a car driving in a straight line
Distance (x)
Distance is measured in meters (m)
Time (t)
Time is measured in seconds (s)
Speed (v)
Units: m/s
Defined to be the distance covered in the elapsed time

3. When driving the distance above, are you always going 26.8 m/s?
Example
The distance from Dayton to Columbus is 114,424 m. If it takes you 71.1 minutes to drive from one to the other, what is your speed?
When driving the distance above, are you always going 26.8 m/s?
The speed found is the average speed
Average speed is the distance covered over the change in time
x = 114,424 m
t = 71.1 min.
t = 4266 s
Δ stands for “change in”

4. Vectors

5. Vectors Distance and speed are only half of what physicists need
Distance and speed are examples of scalars
A scalar is something that can be described as a single number
Examples: 20°C or 55 mph
Vectors have both a magnitude (number) and direction
When typed, a vector is written as a bold, capital letter
When handwritten, a vector is a letter with an arrow over it
Remember this for your lab reports!!!
Typed Vectors
Handwritten Vectors

6. Direction of Vectors
In physics, a negative sign is used to indicated direction
Specifically, it is used to indicate that the vector is in the opposite direction from a vector that is positive
In each problem, you can choose which direction is positive
Right is typically positive on the x-axis
Up is typically positive on the y-axis
These can change though depending on your needs
Vector K as a length of 2 in the direction of 90°
Vector L as a length of 3 in the opposite direction of 90°, which is what?

7. Drawing Vectors
By convention, a vector is an arrow pointing towards the direction specified with the length of the arrow representing the magnitude B A
As you can see, A is drawn so that it is twice as long as B and half as long as C. C

8. A vector will have a set direction and magnitude, it will not have a set starting and ending point
This means that I can move the vectors wherever I want to as long as I keep the magnitude and direction the same
All of the vectors below are the same, since I have kept the same magnitude and direction A A A A A A

9. Vector Addition Graphically Adding Vectors Use the Head-to-Tail method
Line up the vectors head to tail, and the resultant is the sum of the vectors
The resultant vector is the vector that is drawn from the tail of the first vector to the head of the last vector A C R B B C A

10. Example C B I have three vectors I want to add graphically A A
I am going to start by moving A, then B, then C B
For the first vector, it doesn’t matter where you start
Notice how the second vector I used started at the head of the first vector
The third vector starts at the head of the second C

11. A
The sum of these three vectors is the straight line distance from where you started to where you ended
The sum is also called the resultant R R B C C
I could have just as easily have moved C then B then A
The sum is the same (or R has the same length and direction)
Vector addition, just like regular addition, is commutative R B A

12. Adding Parallel Vectors
Parallel vectors lie on the same line
Both sets of vectors below are parallel
To add parallel vectors, add the magnitudes of the vectors
The direction of the resultant will be in the direction of the original vectors R

13. Examples
A has a magnitude of 10 and is oriented along the positive x-axis. B has a magnitude of 17 and is also along the positive x-axis. What is A + B?
A + B = 27 at 0°
C has a magnitude of 8 and is oriented along the positive x-axis. D has a magnitude of 19 and is along the negative x-axis. What is C + D?
C + D = 11 at 180 ° R A B R C D

14. Adding Perpendicular Vectors
Perpendicular vectors lie at right angles to each other
To find the magnitude of the resultant, use the Pythagorean Theorem
Why?
When the vectors are added graphically and the resultant is drawn, they form a triangle where the resultant is the hypotenuse
To find the direction of the resultant, use a trig function, either sine, cosine or tangent R θ

15. Examples
A has a magnitude of +15 along the x-axis. B has a magnitude of +17 along the y-axis. What is A + B?
A + B = at 48°above the positive x-axis
C has a magnitude of -3 along the x-axis. D has a magnitude of +4 along the y-axis. What is C + D?
C + D = 5 at 53°above the negative x-axis R B A R B A

16. Resolving Vectors When you resolve a vector, you are taking it apart
Any vector can be resolved into two components
The component vectors, when added, will give you the original vector
One component will be a vector on the x-axis
The other will be along the y-axis
This process is the opposite of adding perpendicular vectors
It is also the same as giving you the hypotenuse of a triangle and asking you to find the length of the sides R B
R has two components, A and B
The components are graphically added to show that they add up to R A

17. Examples
A has a magnitude 12 at an angle of 20°from the positive x-axis. Find the components of A.
Ax =
Ay = +4.10 A
First thing to do is draw out the vector
Then, I would draw in the components and label them
I typically use the subscripts x and y for the components
From here you just need to solve for the two sides of the triangle A Ay Ax

18. Linear Motion

19. Displacement (x) Vector form of distance
Units: meters
Magnitude of the displacement is the straight line distance from the initial to final positions
Vector points from the initial to final positions
xf and xo are the final and initial positions
In the picture, the distance traveled in in black and the displacement is in red.
Can you ever travel a distance and end up with a displacement = 0?

20. Velocity (v) Vector form of speed
Units: meters per second
The magnitude of the velocity is the speed of the object
In other words, the rate that you are changing your position
That is why the units are distance per time
Direction of the velocity is the direction of the motion
Δt = “change in time”
From here on in, whenever you see a “t” for time it really stands for “Δt”.

21. Instantaneous Velocity
The average velocity is how fast you were going the entire trip
Requires simple algebra to calculate
The instantaneous velocity is how fast you are going at that moment
Requires calculus to calculate
Unfortunately, we will not be calculating the instantaneous velocity of any objects in this class
Instantaneous velocities are not easy to calculate, but they are easy to measure. What would you use to measure it?

22. Acceleration (a) Defined to be a change in velocity
Units: meters per second per second
m/s2
Change in velocity is a change in speed or a change in direction or both
That means that accelerations occur when you change your speed or direction or both
Change in speed means speeding up or slowing down
The word deceleration means “negative acceleration”
The magnitude of the acceleration is the rate that the speed is changing
That is why the units are speed per time

23. Average acceleration is your change in speed over the elapsed time
The acceleration vector points in the direction of motion if the object is speeding up
This vector is positive
The acceleration vector points in the opposite direction of motion if the object is slowing down
This vector is negative
Average acceleration is your change in speed over the elapsed time

24. Instantaneous Acceleration
How much you are accelerating at that moment
Once again, this requires calculus to calculate
We won’t be calculating instantaneous accelerations either in this course

25. Kinematics Equations
These equations are used to describe motion under a constant acceleration
Constant acceleration means that the acceleration is not changing
There are four kinematics equations
We will use these four equations to help us start to describe motion
There are five kinematics variables
x = displacement
vo = initial velocity
v = final velocity
a = acceleration
t = elapsed time

26. Start with the definition of acceleration
Since “Δ” stands for “Change in”, substitute the change in velocity for “Δv”
Solve the above equation for the final velocity

27. Start with the definition of average velocity
Since the average velocity can also be found by averaging the final and initial velocity, substitute this for “v”
Solve the above equation for distance

28. Take the first equation we derived and substitute it in for “v” in the second equation
Simplify the above equation

29. Solve the first equation we derived for “t”
Substitute this equation into the second one we derived
Simplify the above equation and then solve for “v2”

30. Summary of Kinematics Equations
While I will never have you memorize equations, it would be worth your while to learn these. We will be using them ALL year.

31. Free Fall
Free Fall occurs when gravity is the only influence on an object
Any falling object is under a constant acceleration
Since it is under a constant acceleration the kinematics equations apply
The acceleration due to gravity (g) is 9.8 m/s2 near the surface of the Earth
If you get “far” away from the surface of the Earth this value will change
You need to memorize this number

32. The acceleration due to gravity vector points towards the surface of the object
Because gravity always accelerates objects down

33. What is the velocity of a falling object dependent on?
To find out, go the first kinematic equation and plug in “g” for the acceleration
The velocity of a falling object on Earth is dependent on the time it has fallen and the initial velocity
That’s it and nothing more
Notice that the velocity of an object does not depend on the mass
In other words, objects of different mass fall at the same rate.
We can test this on the Earth, but to get an accurate test, we must go to the Moon
This is because the Moon has no air and no air resistance

34. Astronaut David Scott of Apollo 15
He is testing whether objects fall at the same rate in the absence of an atmosphere

38. Physics