1. Kinematics
in One Dimension
Chapter 2

2. Kinematics describes motion in terms of
Kinema is Greek for “motion”
Kinematics → Describing motion
Kinematics describes motion in terms of
position
displacement
velocity (average and instantaneous)
acceleration
Dynamics
Describing how interactions can change motion
Mechanics
Kinematics and dynamics.

3. Position x is the location of the object
initial position
final position
positive x direction
x axis at 0°
For one-dimensional motion, plus (+) and minus (-) are used to specify the vector direction. Plus (+) is at 0° along the positive x axis and minus (-) is at 180° along the negative x axis.

4. Displacement Δx is the change in position
Change Δ always means (final value) – (initial value)

5. Displacement is +5 m Example: Displacement in the positive x direction
Displacement is the change
initial
final
Displacement is +5 m

6. Example: Displacement in the negative x direction
initial
final
Displacement is the change
Displacement is - 5 m

7. Velocity is the displacement in one second.
Average velocity is the displacement divided by the elapsed
time.
v "bar" means average
SI units for velocity: meters per second

8. Example 2 The World’s Fastest Jet-Engine Car
Andy Green in the car ThrustSSC set a world record of
341.1 m/s in The driver makes two runs through the course, one in each direction, to adjust for wind effects. Determine the average velocity for each run.
Vector directions are shown with +/- signs.

9. Ignore the textbook definition of speed
In this course, speed will mean the magnitude of the velocity.
SI units for speed: meters per second

10. Instantaneous velocity is the average velocity for a very short time interval.
Average velocity is the average during some time interval.
Instantaneous velocity is the velocity at a specific moment.

11. Average acceleration is the velocity change in one second.
"bar" means average

12. Example 3 Increasing velocity: Determine the average acceleration.
v and a are in the same direction so v increases

13. Acceleration is the velocity change each second.
v and a are in the same direction so v increases

14. Acceleration is the velocity change each second
v and a are in opposite directions so v decreases

15. Graphical representation of kinematic quantities
Position graph with a constant slope

16. Position graph with three different slopes
Velocity = Slope

17. Position graph with a variable slope
Velocity = Slope

18. Velocity graph with a constant slope

19. Your success in physics critically depends on your ability to correctly distinguish and correctly use technical terms as the terms are defined in physics.
velocity and acceleration are different physical quantities with different units
velocity is position change each second
acceleration is velocity change each second
velocity is different from velocity change

20. Five kinematic variables
1. position (at time t), x
2. initial velocity (at time t=0), vo
3. acceleration (constant), a
4. final velocity (at time t), v
5. final time, t
The kinematic equations relate these 5 variables for situations where the acceleration is constant and allow us to solve for two of these quantities that are unknown.

21. Five kinematic equations for constant acceleration cases
Position equations
Velocity equations
Obtained from the other equations by eliminating t.
Use these equations to solve for any 2 missing variables.
You need to know 3 of the 5 variables to find the missing 2 !
Derivation of the kinematic equations is shown at the end of the slides.

22. Success Strategy [ Very important steps !! ]
1. Make a drawing. 2. Draw direction arrows. (displacement, velocity, acceleration) 3. Label the positive (+) and negative (-) directions.
4. Label all know quantities.
5. Use a table to organize known values with correct units.
6. Make sure you have three of the five kinematic quantities.
7. Select the appropriate equations and solve.
There may be two possible answers because quadratic equations have 2 roots.
For multi-segmented motion, remember that the final velocity of one segment is the initial velocity for the next segment.

23. x a v vo t Example: Find the final position x m s ? 8s
same directions so v increases
Find the final position x x a v vo t m s ? 8s

24. x a v vo t m s ? 8s

25. x a v vo t m s ? Example 6 Airplane take-off Find the final position x
same directions so v increases
Example 6 Airplane take-off
Find the final position x x a v vo t m s ?

26. Example 6 Airplane take-offv vo t m s ?

27. Free fall -- motion only influenced by gravity
In the absence of air resistance, all bodies at the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is small compared to the radius of the Earth, then the acceleration remains constant throughout the motion.
This idealized motion is called free-fall and the acceleration
of a freely falling body is called the acceleration due to
gravity. The direction of this acceleration is downward.
The magnitude of the acceleration is called "g".

28. air
no air (vacuum) (not vacuum cleaner)
air resistance influences falling y
downward
y axis positive upward negative downward

29. A stone is dropped from the top of a tall building
A stone is dropped from the top of a tall building. What is the displacement y of the stone after 3 s of free fall? y a v vo t m s ?
-9.8 m/s2
0 m/s
3 s

30. Vertical velocity always momentarily zero at the top .
A coin is tossed upward with an initial speed of 5 m/s. How high does the coin go above its release point? Ignore air resistance.
Only one value is given in the wording of this problem. You are expected to add two values by thinking.
Vertical velocity always momentarily zero at the top . y a v vo t ?
+5 m/s

31. y a v vo t ?
-9.8 m/s2
0 m/s
+5.00 m/s

32. The velocity changes, but does the acceleration change?
Acceleration versus Velocity
There are three parts to the motion of the coin.
On the way up, the vector velocity is upward and the vector acceleration is downward so the velocity change is downward (v gets less upward).
At the top, the vector velocity is momentarily zero and the vector acceleration is downward so the velocity change is downward (v gets more downward).
On the way down, the vector velocity is downward and the acceleration vector is downward so the velocity change is downward (v gets more downward).
The velocity changes, but does the acceleration change?

33. The End

34. Derivation of the kinematic equations

35. multiply both sides by t
Deriving the kinematic equations
We assume the object is at the origin at time to = 0.
and
Velocity definition
multiply both sides by t
position equation
Also, for constant acceleration cases

36. multiply both sides by t
Acceleration definition
multiply both sides by t
Velocity equation

37. another position equation