Physics Study Notes
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Lesson 4 Linear Motion
Change and Motion
a.
A property common to everything in the universe is change.
b.
Change is so important that the fundamental concept of time would be meaningless without it.
c.
Since changes are a result of the motion of material, often at the submicroscopic level, we start
study physics with a study of motion.
d.
The complex motions in our daily experience can be understood as combinations of simple ones.
So, we begin our discussion of motion by trying to describe and understand the simplest kind of
motion.
Motion is Relative
a.
Motion occurs all around us, either visible or invisible. There are motions at microscopic level
such as jostling atoms make heat and sound, flowing electrons makes electricity, vibrating
electrons producing electromagnetic waves.
b.
When we discuss the motion of something, we discuss its motion relative to something else.
c.
Even things that appear to be at rest, they may move at very high speed relative to the sun and
stars.
Speed
a.
Two fundamental notations – space and time – are at the core of our concept of motion.
b.
Definition: Distance covered per unit of time. Speed is a measure of how fast something is
moving. It is the rate at which distance is covered.
c.
Unit: Meters per second (m/s) is the primary unit for speed. Others such as miles per hour (mi/h),
kilometer per hour (km/h), light-years per century are all legitimate units for speed.
d.
Average speed:
v=
d
,
t
t
d = total distance covered (m)
t = time interval (s)
€
e.
Instantaneous
speed: The speed at any instance of an object is called the
€
instantaneous speed. The instantaneous speed of a moving object is equal
to the slope of the tangent line at that moment.
f.
Constant speed: The speed at any instance of an object is constant.
d
slope of the tangent
€
t
v
d
t
t
Velocity
a.
Definition: Velocity is speed in a given direction. Velocity is how fast and in what direction it
moves. Quantities that have both a size (or magnitude) and a direction are called vectors. The
speed is the magnitude of velocity.
Mr. Lin
d
= slope
t
€
v = average speed (m/s)
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v=
d
total dis tan ce cov ered
average speed =
time int erval
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Physics Study Notes
b.
Lesson 4 Linear Motion
Average velocity:
average velocity =
v=
total displacement
time interval
d
,
t
v = average velocity (m/s)
d = total displacement (m)
€
t = time interval (s)
€
c.
Displacement
is a vector quantity; its magnitude is the straight-line distance between the initial
€
and final locations of the object, and its direction is from the initial location to the final location.
d.
Constant velocity: To have a constant velocity requires both constant speed and constant
direction. Motion at constant velocity is motion in a straight line at constant speed.
60 mi/h East
60 mi/h West
Same Speed
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Different Velocity
e.
Changing velocity: Either speed or direction (or both) is changing and then the velocity is
changing. Motion at constant speed can have changing velocity all the time when it moves along a
curved path.
f.
Car example: In a car there are three controls that are used to change the velocity: the gas pedal,
the brake and the steering wheel.
Acceleration
a.
Definition: The rate at which the velocity is changing is called acceleration. It is a measure of
how the velocity is changing with respect to time.
b.
Acceleration:
change of velocity
acceleration =
time int erval
Δv ,
a = acceleration (m/s2)
a=
t
Δv
= slope
t
€
Δv = change of velocity (m/s)
€
c.
a=
v
t
t = time interval (s)
€ acceleration: The acceleration at any instance of an object is constant. In high school
Constant
physics, we only deal with constant acceleration.
v
a
t
t
d.
Acceleration is change: Acceleration applies to changes in direction as well as changes in speed,
i.e., changes in the state of motion. The acceleration applies to increases as well as decreases in
speed. Sometimes, the decrease in speed is also called deceleration, or negative acceleration.
e.
Acceleration is directional: Acceleration, like velocity, is directional. If we change either speed or
direction, or both, we change velocity and we accelerate. When linear (straight-line) motion is
Mr. Lin
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Physics Study Notes
Lesson 4 Linear Motion
considered, it is common to use speed and velocity interchangeably and the acceleration may be
expressed as the rate at which speed changes.
acceleration (along a straight line) =
f.
change in speed
time int erval
Car example: Cars having good acceleration means being able to change velocity quickly and
does not necessarily refer to how fast something is moving.
€
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Free Fall: How Fast
a.
Definition: When there is no air resistance and the gravity is the only thing affecting a falling
object, such motion is called free fall. Elapsed time: The elapsed time is the time that has elapsed,
or passed, since the beginning of the fall.
b.
Acceleration due to gravity (g): The free falling object is experiencing acceleration, i.e., a change
in speed. The value of acceleration (g) is about 10 m/s2. More accurately, g is 9.81 m/s2.
c.
Instantaneous speed: The instantaneous speed of an object falling from rest
v = 0 m/s
3s
instantaneous speed = acceleration x elapsed time
or
v
g = = slope
t
v
v = gt
v = 10 m/s
2s
4s
t
v = 20 m/s
1s
5s
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d.
Initial speed (vi): Whether the object is moving upward or downward and no matter
what the initial speed it is, the acceleration due to gravity is always the same (9.81 m/s2)
the entire time. That means, during each second, the speed or velocity changes by 9.81
m/s (≈ 10 m/s). The final speed (vf), the instantaneous speed, is:
v
g = = slope
t
vf
vf = vi + gt
vi
v = 40 m/s
7s
€
7
v = 30 m/s
6s
0s
t
Free Fall: How Far
a.
Speed and distance: The speed and distance are different. The instantaneous speed and average
speed are different too. The instantaneous speed of 10 m/s at the end of 1 second does not mean
that the object falls 10 m during the first second. But, if the object falls 10 m in the first second,
the average speed of the object is 10 m/s.
b.
Average speed: For any object moving in linear motion with constant acceleration, the average
Δv
speed is:
a=
= slope
v
initial speed + final speed
average speed =
2
or
c.
v=
€
Distance-time formula: Since
t
vf
v
vi
vi + v f
2
€
t
Δv or v = v + gt
a=
f
i
t
€
v=
vi + v f
2
(1)
(2)
€
€
Mr. Lin
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Physics Study Notes
Lesson 4 Linear Motion
v=
Substitute (2) into (3) and we get:
d=v t =(
€
€
Substitute (1) into (4) and we get:
d.
d
, or d = v t
t
d=(
(3)
vi + v f
)t
2
(4)
v i + v i + gt
1
) t = v i t + gt 2
2
2
€
Free fall: Since vi = 0, the distance-time formula becomes
€
1
1
d = 0 + gt = gt
2
2
2
2
The falling distance of an object is depends on time squared, so, the graph of the distance-versetime is a parabolic. If g = 10 m/s2, the falling distance is shown in the table:
€
Elapsed Time (s)
0
1
2
3
4
5
6
:
t
e.
Distance-Speed formula: Since
Distance Fallen (m)
0
5
20
45
80
125
180
:
½gt2
or
vf = vi + gt
d
t
g=
v −v
t
f
(1)
i
1
d = v t + gt
2
v €− v
1 v −v
d=v (
) + g(
)
g
2
g
Substitute (1) into (2) and we get:
€
vv −v
(v − v )
=(
)+
g
2g
2v v − 2v + v − 2v v + v
=
2g
v −v
=
2g
(2)
2
i
f
i
f
2
i
i
2
i
f
i
f
2
i
f
i
2
i
2
f
i
f
2
i
2
f
Rearrange (3) and we get:
(3)
2
i
2
2
v = v + 2gd
f
i
€
€
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Graphs of Motion
Mr. Lin
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Physics Study Notes
a.
Lesson 4 Linear Motion
Velocity verse time graph:
Velocity
(m/s)
10
5
0
b.
5
10
15
25 Time (s)
20
Displacement verse time graph:
Displacement
(m/s)
150
100
50
0
c.
5
15
5
10
15
Velocity
Acceleration
Displacement
(s)
(m/s)
(m/s2)
(m)
0
0
1
0.0
1
1
1
0.5
3
3
1
4.5
5
5
1
12.5
6
5
0
17.5
7
5
0
22.5
8
5
0
27.5
9
5
0
32.5
10
5
0
37.5
11
5
0
42.5
12
5
0
47.5
13
7
2
53.5
14
9
2
61.5
15
11
2
71.5
16
11
0
82.5
17
11
0
93.5
18
11
0
104.5
19
10.5
-0.5
115.25
20
10
-0.5
125.5
21
9.5
-0.5
135.25
22
9
-0.5
144.5
23
9
0
153.5
24
9
0
162.5
25
6
-3
170.0
26
3
-3
174.5
27
0
-3
176.0
25 Time (s)
20
d.
The slope of the velocity-time graph is equal to the acceleration. The area under the velocitytime graph is equal to the displacement.
e.
For a freely falling object, the acceleration verse time is a constant, the speed verse time is a
direct proportion, and the distance verse time is a parabola.
f.
Hang time: When athletes jump up in the air, the time duration they stay in the air is called hang
time.
Summary of Formulas
v=
d
t
a=
Δv
t
1
d = v t + gt
2
vf = vi + gt
v=
€
€
vi + v f
2
2
2
v = v + 2gd
2
f
i
€
25 Time (s)
20
Acceleration verse time graph:
Acceleration
(m/s2)
3
2
1
0
-1
-2
-3
9
10
Time
i
€
€
Mr. Lin
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Physics Study Notes
Lesson 4 Linear Motion
10 Air Resistance and Falling Objects
a.
Air resistance causes the differences of falling between feather and coin. When the falling is tested
in the vacuum tube, both of them reach the ground at the same time.
11 Linear Motion Example Problems
For all the following problems, assume all the surfaces are frictionless and the air resistance can be
neglected. The value of g can be approximated to 10 m/s2 for simplicity.
a.
James throws a tennis ball upward at 10 m/s, (a) how high will the ball reach? (b) How long will it
take for the ball to fall back to him? (c) What’s the speed of the ball when it falls back?
b.
Mr. Lin is falling into the ocean from a cliff of 1000-meter height, a) how long will it take for Mr.
Lin to reach the surface of water? b) What is Mr. Lin’s speed right before he falls into the water?
c.
Bogy found a hazard road condition 100-meter in front of him. If his car is moving at 10 m/s, (a)
what’s the minimum constant acceleration required to avoid the danger? (b) How long will it take
for his car to be fully stopped under such acceleration?
d.
Thomas’s car moves at 20 m/s while Shalin’s moves at 40 m/s. Both cars are designed to
decelerate at 5 m/s2 when the brakes are engaged. a) What’re the skid distances for both cars? b)
Compare their speeds and skid distances. What kind of conclusion can you draw?
e.
Anna is throwing an object upward from the edge of a 120-meter tall building at 10 m/s and falls
to the ground. (a) How long will it take for the ball to hit the ground? (b) What’s the speed of the
ball before hitting the ground?
f.
If an over-speed car is moving at constant speed of 40 m/s and passing a police car at rest. If the
police starts chasing while it passing and the police car can keep accelerating at 4 m/s2, (a) how
long will it take for the police to catch the over-speed car? (b) How far will it take for the police to
reach the over-speed car? (c) What’s the instantaneous speed of the police car when it catches the
over-speed car?
Mr. Lin
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Physics Study Notes
Lesson 4 Linear Motion
g.
Mary Jane freely falls from the top of a skyscraper of 550-meter height for 5 seconds while
Spiderman is trying to rescue her. If Spiderman jumps from the top of the same building with
initial speed v downward, and he needs to catch her at least 50 m above the ground to ensure her
safety, a) what is the minimum value of v required to finish this task? b) What are the
instantaneous speeds of Mary Jane and Spiderman respectively when they meet each other in the
air?
h.
Victoria slides down a 1000-meter long slope of 30o freely from the top. a) How long will it take
for him to reach the bottom of the slope? b) What’s his speed when he reaches the bottom?
i.
Ball A is freely falling from the top of the building. After ball A falls for a meters, ball B starts
falling freely from a certain floor of the building which is b meters below the top of the building.
If ball A and ball B both reach the ground at the same time, what is the height of the building in
terms of a and b?
Mr. Lin
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