1. Mechanics: Motion in One Dimension x dx Notes by: Ted Vittitoedt x t
Notes by: Ted Vittitoe
Kyle Curry
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2. Motion in One Dimension
Sections
2-01 Displacement
2-02 Velocity
2-03 Acceleration
2-04 Motion Diagrams
2-05 One Dimensional Motion
with Constant Acceleration
2-06 Freely Falling Objects
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3. An inside observer does not feel he/she is moving.
Frame of Reference
A measure of distance, speed, and acceleration are always made in some type of Frame of Reference.
An inside observer does not feel he/she is moving.
An outside observer sees the person inside the train moving at the same velocity as the train
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4. Vectors vs. Scalars
A VECTOR QUANTITY requires both a direction and a magnitude to describe.
A SCALAR QUANTITY is described only by magnitude.
Example:
Velocity (a vector) of a track runner may be described as “13 m/s toward the finish line” while her speed (a scalar) would be described as just “13 m/s.”
Vector
Scalar
Velocity
Speed
Acceleration
Energy (Potential, Kinetic)
Displacement
Mass, Density
Forces, Fields
Time, Volume
Torque, Momentum
Temperature

5. A particle is a point-like mass having infinitesimal
Displacement v 
In the study of kinematics (how do objects move), we consider a moving object as a particle.
A particle is a point-like mass having infinitesimal
size and a finite mass.
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6. Displacement
The displacement of a particle is defined as its change in position (where the object is currently located). x
(m) -6 -4 -2 2 4 6
Dx = x - xo
= 6 m - 2 m
= 4 m
x = Where the object is on a coordinate plane
xo = Where the object is
Note: Displacement to the right is positive
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7. = -6 m - 6 m Displacement The displacement of a particle is defined
as its change in position. x
(m) -6 -4 -2 2 4 6
Dx = x - xo
= -6 m - 6 m
= -12 m
Note: Displacement to the left is negative
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8. = (2 m) - (-6 m) Displacement
The displacement of a particle is defined
as its change in position. x
(m) -6 -4 -2 2 4 6
Dx = x - xo
= (2 m) - (-6 m)
= 8 m
Note: Displacement to the right is positive
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9. A student walks 70 m East, then walks 30 m West.
Displacement
A student walks 70 m East, then walks 30 m West.
What is the magnitude of the students net displacement?
A) 30 m
B) 40 m
C) 70 m
D) 100 m
East
West
70 m
30 m
Displacement
40 m
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10. Velocity is represented displacement-time graph
Average velocity:
Total Displacement ÷ Total Time Interval
The average velocity of a particle is defined as x x1 x2 t1 t2 Dx Dt
Velocity is represented
by the slope on a
displacement-time graph t
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11. Always a positive value (Magnitude of Average Velocity)
Average speed:
Always a positive value (Magnitude of Average Velocity)
The average speed of a particle is defined as
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12. Instantaneous velocity displacement-time graph
The instantaneous velocity v, equals the limiting value
of the ratio x t Dx Dt
Instantaneous velocity
is represented by
the slope of a
displacement-time graph
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13. The instantaneous speed of a particle is defined
Velocity
Instantaneous speed
The instantaneous speed of a particle is defined
as the magnitude of its instantaneous velocity.
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14. Acceleration is represented
Average acceleration
The average acceleration of a particle is defined
as the change in velocity Dvx divided by the time
interval Dt during which that change occurred. v v1 v2 t1 t2 Dv Dt
Acceleration is represented
by the slope on a
velocity-time graph t
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15. A new car manufacturer advertises that their car can
Acceleration
A new car manufacturer advertises that their car can
go "from zero to sixty in 8 s". This is a description of
A) instantaneous acceleration.
B) average speed.
C) instantaneous speed.
D) average acceleration.
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16. Instantaneous acceleration
The instantaneous acceleration equals the derivative
of the velocity with respect to time v t Dv Dt
Instantaneous acceleration
is represented by
the slope of a
velocity-time graph
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17. Acceleration: Deceleration
When an object is slowing down.
This does NOT mean that acceleration is negative
V0 = 15.0 m/s
V = 5.0 m/s
t0 = 0.0 s
t = 5.0 s a
Negative X Direction

18. Acceleration: Acceleration (Negative Direction)
V = m/s
V0 = m/s
t = 5.0 s
t0 = 0.0 s a
Negative X Direction Direction

19. Acceleration: Acceleration Positive Direction X Direction
V = m/s
V0 = m/s
t = 5.0 s
t0 = 0.0 s a
Positive Direction X Direction

20. A) traveling at 1.5 m/s in every second.
Acceleration
A moving car experiences a constant acceleration of 1.5 m/s2. This means the car is
A) traveling at 1.5 m/s in every second.
B) changing its velocity by 1.5 m/s.
C) increasing its velocity by 1.5 m/s in every second.
D) increases its displacement by 1.50 m each second.
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21. (a) A car must always have an acceleration in the same
Quick Quiz (page 32)
True or False?
(a) A car must always have an acceleration in the same
direction as its velocity
False
(b) It’s possible for a slowing car to have
a positive acceleration
True
(c) An object with constant nonzero acceleration can never
stop and stay stopped.
True
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22. Formulas for Motion in One Dimension
These are the formulas that we will be using to solve problems of 1-D Motion in this course and on the AP Physics B Exam.

23. Motion Diagrams
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
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24. Motion Diagrams (con’t)
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
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25. Motion Diagrams (con’t)
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
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26. Motion Diagrams (con’t)
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
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27. Motion Diagrams (con’t)
The displacement versus time for a certain particle moving along the x axis is shown below. Find the average velocity in the time intervals
(a) 0 to 2 s
(b) 0 to 4 s
(c) 2 s to 4 s
(d) 4 s to 7 s
(e) 0 to 8 s.
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28. An object starts from rest and moves with constant acceleration.
Motion Diagrams
An object starts from rest and moves with constant acceleration. 4 8 12 16 20 24 28
(s) x 4 8 12 16 20 24 28
(m) 1 2 3 t 5
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29. (s) x 4 8 12 16 20 24 28 (m) 1 2 3 t 5 Motion Diagrams 1 2 3 4 5 t (s)10 v
(m/s)
Displacement
25 m
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30. Displacement, velocity and acceleration graphs x
Motion Diagrams
Displacement, velocity
and acceleration graphs x
The slope of a displacement-time
graph represents velocity t v
The slope of a velocity-time
graph represents acceleration t a t
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31. Displacement, velocity and acceleration graphs x
Motion Diagrams
Displacement, velocity
and acceleration graphs x
The area under a velocity-time
graph represents displacement. Dx t v
The area under an acceleration-time
graph represents change in velocity. Dv t a Dt t
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32. The slope of a position versus time graph gives A) position.
Motion Diagrams
The slope of a position versus time graph gives
A) position.
B) velocity.
C) acceleration.
D) displacement.
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33. The slope of a velocity versus time graph gives A) position.
Motion Diagrams
The slope of a velocity versus time graph gives
A) position.
B) velocity
C) acceleration
D) displacement
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34. One Dimensional Motion with Constant Acceleration
Definitions of velocity and acceleration
Average velocity
Average acceleration
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35. One Dimensional Motion with Constant Acceleration
For constant acceleration
An object moving with an initial velocity vo undergoes
a constant acceleration a for a time t. Find the final velocity. vo ? a
time = 0
time = t
Solution:
Eq 1
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36. One Dimensional Motion with Constant Acceleration
What are we calculating? t a DV
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37. One Dimensional Motion with Constant Acceleration
Objects A and B both start at rest. They both accelerate
at the same rate. However, object B accelerates for twice
the time as object A. What is the final speed of object B
compared to that of object A?
A) the same speed
B) twice as fast
C) three times as fast
D) four times as fast
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38. One Dimensional Motion with Constant Acceleration
For constant acceleration
An object moving with a velocity vo is passing position xo when it undergoes a constant acceleration a for a time t. Find the object’s displacement.
time = 0
time = t xo ? a vo
Solution:
Eq 2
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39. One Dimensional Motion with Constant Acceleration
What are we calculating? t vo v at
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40. One Dimensional Motion with Constant Acceleration
Objects A and B both start at rest. They both accelerate
at the same rate. However, object B accelerates for twice
the time as object A. What is the distance traveled by object B compared to that of object A?
A) the same distance
B) twice as far
C) three times as far
D) four times as far
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41. One Dimensional Motion with Constant Acceleration
Eq 2
Eq 1
Solve Eq 1 for a and sub into Eq 2:
Eq 3
Solve Eq 1 for t and sub into Eq 2:
Eq 4
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42. One Dimensional Motion with Constant Acceleration
When the velocity of an object is zero, must its
acceleration also be zero?
A) no, an object thrown upward will have zero velocity
at its highest point.
B) no, a falling object will have zero velocity after
hitting the ground.
C) yes, if the object is not moving it can not be
accelerating.
D) yes, acceleration implies a changing velocity, it can
not be zero.
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43. Freely Falling Objects
When an object is released from rest and falls in the
absence of air resistance, which of the following is true
concerning its motion?
A) Its acceleration is constant
B) Its velocity is constant.
C) Neither its acceleration nor its velocity is constant.
D) Both its acceleration and its velocity are constant.
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44. (a) How long does it take for the lead car to stop?
Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2.
(a) How long does it take for the lead car to stop?
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45. (b) How far does the lead car travel during the acceleration?
Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2.
(b) How far does the lead car travel during the acceleration?
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46. Problem Alternate Solutions
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47. Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2.
(c) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing car’s minimum negative acceleration so as not to hit the lead car?
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48. (d) How long does it take for the chasing car to stop?
Problem
Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –2.00 m/s2.
(d) How long does it take for the chasing car to stop?
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49. Problem Alternate Solutions
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50. A Cessna aircraft has a lift-off speed of 120 km/h.
Problem
A Cessna aircraft has a lift-off speed of 120 km/h.
(a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 240 m?
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51. A Cessna aircraft has a lift-off speed of 120 km/h.
Problem
A Cessna aircraft has a lift-off speed of 120 km/h.
(b) How long does it take the aircraft to become airborne?
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52. (a) How long did it take the race car to travel this distance?
Problem
A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m (1/4 mile).
(a) How long did it take the race car to travel this distance?
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53. (b) What is the speed of the race car at the end of the run?
Problem
A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m (1/4 mile).
(b) What is the speed of the race car at the end of the run?
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54. Problem
A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise?
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55. A ball is thrown vertically upward with a speed of 25.0 m/s.
Problem
A ball is thrown vertically upward with a speed of 25.0 m/s.
(b) How long does it take to reach its highest point?
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56. A ball is thrown vertically upward with a speed of 25.0 m/s.
Problem
A ball is thrown vertically upward with a speed of 25.0 m/s.
(c) How long does the ball take to hit the ground after it reaches its highest point?
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57. A ball is thrown vertically upward with a speed of 25.0 m/s.
Problem
A ball is thrown vertically upward with a speed of 25.0 m/s.
(d) What is its velocity when it returns to the level from which it started?
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58. Kinematics with Constant Acceleration
Review
Definitions
Average velocity
Average acceleration
Kinematics with Constant Acceleration
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59. Review t x v a t x v a Dt Dv Dx
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60. Problem Solving Skills
Review
Problem Solving Skills
1. Read the problem carefully
2. Sketch the problem
3. Visualize the physical situation
4. Identify the known and unknown quantities
5. Identify appropriate equations
6. Solve the equations
7. Check your answers
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61. END

62. Constant Velocity
Suppose that an object is moving with a constant velocity. Make a statement concerning its acceleration.
A) The acceleration must be constantly increasing.
B) The acceleration must be constantly decreasing.
C) The acceleration must be a constant non-zero value.
D) The acceleration must be equal to zero.

63. Freely Falling Objects
Can an object have increasing speed while the magnitude of its acceleration is decreasing? Support your answer with an example.
A) No, this is impossible because of the way in which
acceleration is defined.
B) No, because if acceleration is decreasing the object will
be slowing down.
C) Yes, and an example would be an object falling in the
absence of air friction.
D) Yes, and an example would be an object released from
rest in the presence of air friction.

64. Freely Falling Objects
Suppose a ball is thrown straight up. Make a statement about the velocity and the acceleration when the ball reaches the highest point.
A) Both its velocity and its acceleration are zero.
B) Its velocity is zero and its acceleration is not zero.
C) Its velocity is not zero and its acceleration is zero.
D) Neither its velocity nor its acceleration is zero.

65. Freely Falling Objects
Ball A is dropped from the top of a building. One second later, ball B is dropped from the same building. As time progresses, the distance between them
A) increases.
B) remains constant.
C) decreases.
D) cannot be determined from the information given.