137
MOTION IN ONE DIMENSION
MOTION IN ONE DIMENSION
Time n. The period during which an action or process, continues“Webster's New Practical School Dictionary.
“Time is a river flowing into nowhere...” Steve Winwood, singer.
“Time - The measure of duration." Lectric Law Library's Lexicon, http://www.lectlaw.com/def2/t088.htm
"Time is about things wearing out," John Gribbin, astrophysicist.
“Time - How long an event lasts.“ Fundamentals of Physics, Halliday & Resnick, & Walker.
“Time, an endless song,“William Butler Yeats, Anglo-poet and dramatist.
"Time is the school in which we learn, the fire in which we burn.“Delmore Schwartz, American poet and writer
"I am mighty, world-destroying Time,"Bhavagad Gita, 250 B. C., Hindu Scripture
“Time is the most valuable thing a man can spend.” Theophrastus, 200 B. C., Greek philospopher
Let’s talk the same language…
If all the electronic
and mechanical
watches, timers,
and clocks in the
world went on
strike and
stopped working at
the same time how
could you tell time?
7-time winner of the Tour de
France. World Record holder
Avg. Human rest
heart rate:
65 - 80 beats per minute
Lance Armstrong
rest heart rate:
32beats per minute
DIGITAL VS. ANALOG
In order to observe something, it has to be
slowed down/chopped up to be observed
Scientists
are in the
business of
1
2
3
4
5
6
7
8
9
10
11
12
How many frames per second to
trick the human eye?
24 – 28 frames per second
G.U.E.S.S. Method
G.ivens. Identify your given(s)
and name them.
U.nknown. Identify what variable(s) you
are trying to find.
E.quation. Identify your working
equation(s).
S.olve. Solve for your unknown(s).
S.ubstitute. Substitute your given(s).
Snow White was the first feature-length
animated movie. Suppose it ran for an
hour and fifteen minutes. How many
individual images had to be made to
make a smooth looking movie? Use
the G.U.E.S.S. method.
G.ivens: 24 frames per second, 1.25
hours
U.nknowns: no. of images in a whole
movie
E.quation:
No. of frames = 𝐟𝐫𝐚𝐦𝐞𝐬
𝐬𝐞𝐜𝐨𝐧𝐝×
𝐬𝐞𝐜𝐨𝐧𝐝𝐬
𝐡𝐫×
𝐡𝐫𝐬
𝐰𝐡𝐨𝐥𝐞 𝐦𝐨𝐯𝐢𝐞
S.olve: Its already solved
S.ubstitute:
No. of frames =𝟐𝟒 𝐟𝐫𝐚𝐦𝐞𝐬
𝐬𝐞𝐜𝐨𝐧𝐝×
𝟑𝟔𝟎𝟎 𝐬𝐞𝐜𝐨𝐧𝐝𝐬
𝐡𝐫×
𝟏.𝟐𝟓 𝐡𝐫𝐬
𝐦𝐨𝐯𝐢𝐞
No. of frames = 108,000
3
4
5
6
7
8
P.O.D. 1: If the average TV camera frame rate is 24 frames per
second and a Boeing 767 is 61.3 meters long, how fast was the plane
moving into the Second Twin Tower? Use the G.U.E.S.S. method.
1 spin every
3 seconds
1 spin every
2 seconds
1 spin every
1 seconds
1 spin every
0.5 seconds
1 spin every
0.1 seconds
P.O.D. 3: Answer the following questions
1. This stop action photograph lets us see that
even though the coin is flipping round and
round, it still follows what kind of path?
2. This stop action photograph
let’s us see that if you were
to plot/graph the girl’s
height vs time, you would
get what kind of shape?
3. How might Mrs. Knight, the SMHS Diamond
Line coach, benefit from having stop action
photography?
High resolution imagehttp://www.washingtonpost.com/wp-srv/special/local/inauguration-2013/pano/d/
Why do we know that time
travel is impossible?
Cuz nobody done did it yet…
• TIME DILATION: It is therelativistic effect that makes movingclocks run slower than stationaryones.
• It is time dilation that allowsastronauts in science fiction to travelvast distances at incredible speedsand return to earth only to find thatthousands of years have passed.
“The length of a minute depends on whichside of the bathroom door you are on”
How can you measure the reaction time (in seconds) of a classmate with a ruler (in cm)?
From Unit 1, we foundthat when an object fallsover time, the relationshipbetween the distancefallen and the time elapsed is
a. Directly proportional
b. Inversely proportional
c. Not related.
From Unit 1, we foundthat when somethingfalls, in the relationshipbetween the distancefallen and the time elapsed, which was thedependent variable and which was theindependent variable?
From Unit 1, we found that when something falls, in the relationship between the distance fallen and the
time elapsed, which table below most accuratelyrepresents the distance vs. time data?
A. 1 50
2 100
3 150
B. 1 5
2 20
3 45
C. 1 150
2 100
3 50
D. 1 36
2 9
3 4
From Unit 1, we looked at the relationship between the distancefallen by an object and the time elapsed. Which graph below
best represents that relationship?
A B
C
D E
From Unit 1, we found that whensomething falls, in the relationship
between the distance fallen and the time elapsed, which formula below best
represents the relationship?
y = kx y = kx2
y = k𝟏
𝒙y = k
𝟏
𝒙𝟐
y = k 𝒙
y/5 = t
y = 5t2
Distance fallen
(in meters) Proportionality
constantTime
(in sec)
y = 5t2
y/5 = t2
5 5
VERIFY YOUR REACTION TIME,
using the ruler method
and the formula y = ½gt2 .
P. O. D. 4: You are driving at 65 miles per hour
(100 km/hr) on a road and are tailgating another
car. The driver of the car sees an old lady
crossing the road and she hits her brakes and
comes to a complete stop. When she hits the
brakes you are 8 meters away from the car.
With the reaction time that you found
earlier, will you crash into the car before
activating your brakes?
1) Distance:
2) Displacement:
3) Position:
Motion can be measured in 3 ways:
Motion can be measured in 3 ways
1) Distance: How far something has moved
since it left its point of origin, the path
matters
2) Displacement:
3) Position:
Motion can be measured in 3 ways
1) Distance: How far something has
moved since it left its point of origin,
the path matters
2) Displacement: The straight line
distance from the original position to the
final position and direction.
3) Position:
Motion can be measured in 3 ways
1) Distance: How far something has
moved since it left its point of origin,
the path matters
2) Displacement: The straight line
distance from the original position to
the final position and direction.
3) Position: The location of the object
Suppose a kick-off returner runs from the end-
zone of a football field, to the fifty yard line,
then back to the twenty yard line because he
forgot the football.
a) What is the distance the kick-off
returner ran?
b) What is the displacement the
person ran?
c) What is the person’s position?
50 + 30 = 80 yards
20 yards
The 20 yard line
A race car goes once around a
track that has a radius of 900 meters.
The driver stops exactly at the
starting point.
900 m
a) What is the distance the race car drove?
Distance = Circumference = 2πr
Distance = 2π(900 m)
900 m
Distance = 1800π m
Displacement?
Position?
0 m
0 m
POD 1:
Distance vs. Displacement What is the distance (in yards) travelled by the football player
in the following diagram?
What is the displacement (in yards) of the football player?
0 10 20 30 40 50 40 30 20 10 0
RA
TT
LE
RS S
LU
GS
Rate – a quantity divided by another quantity
EX: dogs/cat, cell phones/student, kilos/serving
Let’s talk the same language…
𝒓 =𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝟏
𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝟐
The number that you think
about with a rate is the
proportionality constant.
EX: 2 cats per dog, 1 boy per
girl, 0.5 kilos of BBQ per
serving, etc.
Average Speed – a rate that is distance over time
Let’s talk the same language…
EX: km/hr, meters/second, feet/minute
𝒔𝒑𝒆𝒆𝒅𝒂𝒗𝒈 = 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆
𝒕𝒊𝒎𝒆
𝒔𝒂𝒗𝒈 =𝒅
𝒕=𝒅𝒇− 𝒅𝒊𝒕𝒇− 𝒕𝒊
Velocity – a rate that is displacement over
time but has direction
Let’s talk the same language…
EX: km/hr north, meters/second to the left,
feet/minute at an angle of 32º
𝒗𝒂𝒗𝒈 =𝒙𝒕
=𝒙𝟐−𝒙𝟏
𝒕𝟐−𝒕𝟏
𝒗𝒂𝒗𝒈 =𝒙𝒕
=𝒙𝟐−𝒙𝟏
𝒕𝟐−𝒕𝟏
Let’s talk the same language…
Did you learn something new today? Naaah! Actually the average
speed/velocity is just the slope of the line.
𝒎 =𝒚𝒙
=𝒚𝟐−𝒚𝟏
𝒙𝟐−𝒙𝟏
www.PaveYourLane.com
Name: Katie Visco
Talent: Running
Feat: Ran across
the U. S.
“Use your passion to serve humanity.”
In 2009, Pave Your Lane’s
Founder, Katie Visco, ran
solo across America, from
Boston to San Diego to
spread this campaign. Visco
became the 2nd youngest
and 13th woman overall to
run the 3,132 miles coast to
coast.
www.PaveYourLane.com
Distance:
3132 milesAvg. Time:
4 hours a day for
4 months.
“Use your passion to serve humanity.”
What was Katie’s avg. speed?
Find the skeletal walker’s average speed (in yds/min)
Find the skeletal walker’s average velocity (in yds/min.)
sta
rt
finis
h
POD 2:
Fastest man alive22-year old Usain
Bolt of Jamaica, is
officially the
world's fastest
man, clocking an
incredible 100 m
time of
9.58 s at the 2009 World
Championship.
100 m/9.58 s = 10.4 m/s
He is the first man to get below
the 9.7 second mark and has
done it more than once.
How fast is that?
• .
Fastest land animal alive
Cheetahs have been clocked at 100 km/hr for
brief spurts.
This amounts to 27.7 m/s
0 75 150 225 300 375time (in sec.)
600
450
300
150
DC
B
A
The graph below shows the distance a jogger ran in a certain
time t.
How far did the jogger run in 375 s?
The graph below shows the distance a jogger ran in a certain time t.
In which time interval (A, B, C, or D) did the
jogger cover the most distance?
0 75 150 225 300 375time (in sec.)
600
450
300
150
DC
B
A
The graph below shows the distance a jogger ran in a certain time t.
How much time (in minutes) did the
jogger rest?
0 75 150 225 300 375time (in sec.)
600
450
300
150
DC
B
A
The graph below shows the distance a jogger ran in a certain time t.
What was the jogger's velocity in the
first time interval "A" (in m/s)?
0 75 150 225 300 375time (in sec.)
600
450
300
150
DC
B
A
The graph below shows the distance a jogger ran in a certain
time t.
In what time interval(s) did the jogger
accelerate?
0 75 150 225 300 375time (in sec.)
600
450
300
150
DC
B
A
The graph that follows represents the average velocity vs. time of a speedster that is moving at a constant velocity and is exceeding the speed
limit and a police car which spots him and begins to accelerate from rest at a constant rate
to catch him and give him a speeding ticket.
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
What is the speedster’s velocity?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. 30 m/s
What is the police car’s initial velocity?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. 0 m/s
What is the police car’s final velocity?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. 50 m/s
What do you call it when an object
changes velocity?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. acceleration
What is the formula for acceleration?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. Acceleration = 𝐜𝐡𝐚𝐧𝐠𝐞 𝐨𝐟 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲
𝐜𝐡𝐚𝐧𝐠𝐞 𝐨𝐟 𝐭𝐢𝐦𝐞
a = ∆𝒗
∆𝒕 a =
𝒗𝒇−𝒗𝒊
𝒕𝒇−𝒕𝒊
What is the acceleration of the police car?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. a = ∆𝒗
∆𝒕 a =
𝒗𝒇−𝒗𝒊
𝒕𝒇−𝒕𝒊 a =
𝟓𝟎𝒎
𝒔−𝟎
𝒎
𝒔
𝟒 𝒔 −𝟎 𝒔
a = 𝟓𝟎
𝒎
𝒔
𝟒 𝒔 a = 𝟏𝟐. 𝟓 𝒎/𝒔𝟐
How much distance does the speedster travel in 4 seconds?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. G. Velocity is 30 m/s. Time = 4 s.
U. Dist = ?
E. x = vt
S. x = vt
S. x = 30 m/s 4 s = 120 m
x = v t but v is not constant.
If we add the initial and final
velocities vi + vf and divide by 2 we
get an average v.
How much distance does the police car travel in 4 secs?
50
40
30
20
10
0 1 2 3 4 5 6 7
time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. G. vi = 0 m/s, vf = 50 m/s. Time = 4 s.
U. Displacement = x = ?
E. x = ½ (vi + vf)t
S. x = ½ (vi + vf)t
S. x = ½ (0 m/s + 50 m/s) 4 s = 100 m
½ (vi + vf)
Assuming that the police car begins accelerating the instant the speedster passes him, how much time has passed before the police car
has the same velocity as the speedster?
50
40
30
20
10
0 1 2 3 4 5 6 7 time (in seconds)
speedster’s
velocity
police car’s
velocity
vel
oci
ty (
in m
/s)
Ans. G. vip = 0 m/s, vfp = 50 m/s, ap = 12.5 m/s2, vs = 30 m/s
U. t = ?
E. a = ∆𝒗
∆𝒕=
𝒗𝒇−𝒗𝒊
𝒕𝒇−𝒕𝒊
S. 𝐚 =𝒗𝒇−𝒗𝒊
𝒕𝒇−𝒕𝒊 𝒕𝒇 − 𝒕𝒊 𝐚 =
𝒗𝒇−𝒗𝒊
𝒕𝒇−𝒕𝒊 𝒕𝒇 − 𝒕𝒊 𝒂 𝒕𝒇 − 𝒕𝒊 = 𝒗𝒇 − 𝒗𝒊
𝒂 𝒕𝒇 = 𝒗𝒇 𝒂∙𝒕𝒇
𝒂=
𝒗𝒇
𝒂 tf =
𝒗𝒇
𝒂
S. tf = 𝒗𝒇
𝒂=
𝟑𝟎𝒎
𝒔
𝟏𝟐.𝟓 𝒎/𝒔𝟐= 2.4 s
550
500
450
400
350
300
250
200
150
100
50
0 1 2 3 4 5 6 7 8 9 10 11 12 time passed (in minutes)
d
Dista
nce (in
meter
s)
cb
a
a. Find the runner’s average velocity in segment a.
b. In which segment(s) is acceleration happening?
c. In which segment(s) is deceleration happening?
d. How many seconds (NOT MINUTES) does the runner
rest?
e. In what segment does the runner run the most dist.?
POD 3:
Running Man Experiment
c. Find the average speed of your runner over the time interval
d. Find the fastest speedof your runner over your time interval.
time
tota
l dis
t.
EX: d = 4.5t1.43325561
sm7.25
s6.9
m50
t
dv EX:
EX: v = 6.449650245 t0.43325561
v = 14.89 m/s
DIRECTIONS:a. Make a total distance vs. time graph
for your runner’s data. Label your
axes appropriately. Draw a smooth
curve between the points.
b. Derive a formula that describes your
runner’s run as accurately as possible.
The formula will be of the form
d = kt2
STEP 1: Convert the distance PER meter to total
distance:
a. Derive a formula that describes your
runner’s run as accurately as possible.
time
dist. each
meter
(in meters)
0 sec 0 m
1 3.6
2 5.9
time
total dist.
(in meters)
0 sec 0 m
1 0 + 3.6 = 3.6
2 0 + 5.9 + 3.6 = 9.5
3.6
9.5
YOUTUBE
• Thrust Supersonic 1
• Thrust Supersonic 2
• Fastest bicycle 1
• Fastest bicycle 2
You see skid marks appearing after 12 minutes pass.
*Data derived from article in Popular Science, October 1993.
Data worth Big $$$ to the Paparazzi
Fastest man-powered vehicle on the Planet
You see skid marks appearing after 30 hours pass.
**Data derived from article in Ultima Hora Newspaper, 14 October 1997.
Time (in hrs) 0** 1 3 5Total Dist. (in mil) 0 12.717 114.46 317.93
Fastest motorized vehicle on the Planet
Time (in min) 0* 3 6 9Total Dist.(in mil) 0 0.53569 2.1428 4.8212
Displacement formula Velocity formula
x = vot ½at2 v = vo at
HORIZONTAL MOTIONFORMULAS
Where
x = horizontal displacement (in m)
vo= initial velocity (in m/s)
t = time (in s)
v = final velocity (in m/s)
and a = acceleration (in m/s2)
Displacement formula Velocity formula
x = vot ½at2 v = vo at
HORIZONTAL MOTIONFORMULAS
What happens if an object
is not moving initially.
How do the formulas
simplify?
Ans. x = ½a t2 and v = at
HORIZONTAL MOTIONFORMULAS
What happens if an object is
moving at a constant speed and
has no acceleration. How do the
formulas simplify?
x = vot and v = vo
Displacement formula Velocity formula
x = vot ½at2 v = vo at
x= distance, vo = initial velocity, t = time,
v = final velocity, and a = acceleration
Distance
formula
Velocity
formula
Formula x = vot ½at2 v = vo at
MODIFICATION:
+ initial velocity but NO accel.x = vot v = vo
MODIFICATION:
NO initial velocity, but + accel.x = ½at 2 v = at
MODIFICATION:
+ initial velocity & +accel.x = vot ½at2 v = vo at
MODIFICATION:
+ initial velocity & –accel.x = vot ½at2 v = vo at
½at2 at
vot vo
++
−−
HORIZONTAL MOTION FORMULASx = vot,
x = vot ± ½at 2
x = ½at 2
v = vo
v = vo ± at,
v = at,
Starting
Line
time
dis
tan
ce
Is the graph a straight line or a curve?
Ans. A straight line
Does the car have initial velocity?
Ans. YESDoes the car accelerate?
Ans. NO
HORIZONTAL MOTION FORMULASx = vot,
x = vot ± ½at 2
x = ½at 2
v = vo
v = vo ± at,
v = at,
Starting
Line
time
dis
tan
ce
Is the graph a straight line or a curve?
Ans. A curve
Does the car have initial velocity?
Ans. NODoes the car accelerate?
Ans. YES
HORIZONTAL MOTION FORMULASx = vot,
x = vot ± ½at 2
x = ½at 2
v = vo
v = vo ± at,
v = at,
Starting
Line
time
dis
tan
ce
Is the graph a straight line or a curve?
Ans. A curve
Does the car have initial velocity?
Ans. YES
Does the car accelerate?
Ans. YES
|||HORIZONTAL MOTION FORMULAS
x = vo t,
x = vo t ½ at 2
x = ½ at 2
v = vo
v = vo at,
v = at,
Starting
Line
Does the car have initial velocity?
Ans. YES
Does the car accelerate?
Ans. YES, BUT IN A NEGATIVE SENSE
time
dis
tan
ce
Is the graph a straight line or a curve?
Ans. A curve
HORIZONTAL MOTION FORMULASx = vot,
x = vot ± ½at 2
x = ½at 2
v = vo
v = vo ± at,
v = at,
Starting
Line
time
dis
tan
ce
Is the graph a straight line or a curve?
Ans. A straight line
Does the car have initial velocity?
Ans. YES, BUT IN A NEGATIVE DIRECTIONDoes the car accelerate?
Ans. NO
1 sec 1 sec 1 sec1 sec
1 sec 1 sec 1 sec1 sec
0 1 2 3 4 Time (in seconds)
0 1 2 3 4 Time (in seconds)
Distance vs. time
0 1 2 3 4 Time (in seconds)
Distance vs. time
0 1 2 3 4 Time (in seconds)
Distance vs. time
0 1 2 3 4 Time (in seconds)
Distance vs. time
d
d
d
d
0 1 2 3 4 Time (in seconds)
Velocity vs. time
Acceleration
happens…
Find the slope
Acceleration
doesn’t
happen…
slope = 0
Decceleration
happens.
Find the slope
Acceleration
happens
again…
Find the slope
0 1 2 3 4 Time (in seconds)
Acceleration vs. timePositive
Acceleration
happens…
Acceleration
doesn’t happen…
a = 0
Decceleration
happens.
Acceleration happens
again. More than before.
