Motion of a Motion of a

Freely Freely Falling Falling BodyBody

ObjectivesObjectives

►Explain Galileo’s theory of motion Explain Galileo’s theory of motion and Newton’s laws of motion.and Newton’s laws of motion.

►Apply all the equation for motion of Apply all the equation for motion of an object in free fall.an object in free fall.

►To solve problems on free falling To solve problems on free falling bodies and the acceleration of bodies and the acceleration of gravity.gravity.

GalileoGalileo►The remarkable The remarkable

observation that all observation that all free falling objects free falling objects fall at the same rate fall at the same rate was first proposed was first proposed by by Galileo,Galileo, nearly nearly 400 years ago. 400 years ago.

►Galileo conducted experiments Galileo conducted experiments using a ball on an inclined plane to using a ball on an inclined plane to determine the relationship between determine the relationship between the time and distance traveled.the time and distance traveled.

GalileoGalileo►He found that the He found that the

distance depended on the distance depended on the square of the time and square of the time and that the velocity that the velocity increased as the ball increased as the ball moved down the incline. moved down the incline.

►The relationship was the same The relationship was the same regardless of the mass of the ball regardless of the mass of the ball used in the experiment. used in the experiment.

►The story that Galileo The story that Galileo demonstrated his demonstrated his findings by dropping findings by dropping two cannon balls off two cannon balls off the Leaning Tower of the Leaning Tower of Pisa is just a legend. Pisa is just a legend.

►However, if the However, if the experiment had been experiment had been attempted, he would attempted, he would have observed that have observed that one ball hit before the one ball hit before the other! other!

GalileoGalileo

Introduction to Free Introduction to Free FallFall► A free-falling object is an object which is A free-falling object is an object which is

falling under the sole influence of gravity. falling under the sole influence of gravity. ► That is to say that any object which is That is to say that any object which is

moving and being acted upon only be the moving and being acted upon only be the force of gravity is said to be force of gravity is said to be "in a state of "in a state of free fall."free fall."

► This definition of free fall leads to two This definition of free fall leads to two important characteristics about a free-important characteristics about a free-falling object:falling object: Free-falling objects Free-falling objects do not encounter air do not encounter air

resistance. resistance. All free-falling objects (on Earth) All free-falling objects (on Earth)

accelerate downwards at a rate of accelerate downwards at a rate of 9.8 9.8 m/s/s or 32ft/s/sm/s/s or 32ft/s/s

Freely Falling BodyFreely Falling Body► The acceleration of freely falling The acceleration of freely falling

body is so important that physicist body is so important that physicist called it called it acceleration due to gravity. acceleration due to gravity.

► Denoted by letter Denoted by letter gg which is which is equivalent to equivalent to 32ft/sec32ft/sec22 or 9.8m/sec or 9.8m/sec22..

► Meaning in the 1Meaning in the 1stst second, a falling second, a falling body accelerates from a stationary body accelerates from a stationary position to a velocity ofposition to a velocity of 9.8m/sec9.8m/sec2,2, after 2 seconds, the velocity is after 2 seconds, the velocity is doubled to doubled to 19.6m/sec19.6m/sec2 2 after 3 after 3 seconds it triples to seconds it triples to 29.4m/sec29.4m/sec22

► Since accelerating objects are Since accelerating objects are constantly changing their constantly changing their velocity, you can say that the velocity, you can say that the distance traveled divided by the distance traveled divided by the time taken to travel that distance time taken to travel that distance is not a constant value. is not a constant value.

► A falling object for instance A falling object for instance usually accelerates as it falls.usually accelerates as it falls.

► The The fact that the distance which fact that the distance which the object travels every interval the object travels every interval of time is increasing is a sure of time is increasing is a sure sign that the ball is speeding upsign that the ball is speeding up as it falls downward as it falls downward

Freely Falling BodyFreely Falling Body

A simple rule to bear in mindA simple rule to bear in mind► is that all objects is that all objects

(regardless of their mass) (regardless of their mass) experience the same experience the same acceleration when in a state acceleration when in a state of free fall. of free fall.

► When the only force is When the only force is gravity, the acceleration is gravity, the acceleration is the same value for all the same value for all objects. objects.

► On Earth, this acceleration On Earth, this acceleration value is 9.8 m/s/s that it is value is 9.8 m/s/s that it is given a special name - the given a special name - the acceleration of gravity - and acceleration of gravity - and a special symbol - g.a special symbol - g.

NewtonNewton

►Newton's first Newton's first law - the law - the law of law of interactioninteraction states states that that for every for every action there's an action there's an equal and equal and opposite opposite reaction.reaction.

NewtonNewton

►Newton's second Newton's second law - the law - the law of law of accelerationacceleration states states that the that the acceleration of an acceleration of an object is directly object is directly related to the net related to the net force and inversely force and inversely related to its mass.related to its mass.

Newton’s Law of AccelerationNewton’s Law of Acceleration

► Fnet = m * aFnet = m * a

►A=F/mA=F/m

Acceleration depends upon two Acceleration depends upon two factors: force and mass.factors: force and mass.

► The 10-kg elephant The 10-kg elephant obviously has more mass obviously has more mass (or inertia). This increased (or inertia). This increased mass has an inverse effect mass has an inverse effect upon the elephant's upon the elephant's acceleration. acceleration.

► And thus, the direct effect And thus, the direct effect of greater force on the 10-of greater force on the 10-kg elephant is kg elephant is offsetoffset by the by the inverse effect of the greater inverse effect of the greater mass of the 10-kg elephant; mass of the 10-kg elephant; and so each object and so each object accelerates at the same accelerates at the same rate - approximately 10 rate - approximately 10 m/s/s. m/s/s.

One NewtonOne Newton is defined as the amount is defined as the amount of force required to give a 1-kg mass an of force required to give a 1-kg mass an

acceleration of 1 m/s/sacceleration of 1 m/s/s..

Complete the tableComplete the table

Net ForceNet Force(N)(N)

Mass Mass (kg)(kg)

AccelerationAcceleration(m/s/s)(m/s/s)

1.1. 1010 22

2.2. 2020 22

3.3. 2020 44

4.4. 22 55

5.5. 1010 1010

5 m/s/s

10 m/s/s

5 m/s/s

10

1

Free Fall and the Acceleration of Free Fall and the Acceleration of Gravity Gravity

► Free-falling objects are in a state Free-falling objects are in a state of of acceleration. acceleration.

► The velocity of a free-falling The velocity of a free-falling object is changing by object is changing by approximately 10 m/sapproximately 10 m/s every every second. second.

► If dropped from a position of rest, If dropped from a position of rest, the object will be traveling the object will be traveling approximately approximately 10 m/s at the end 10 m/s at the end of the first secondof the first second, approximately , approximately 20 m/s at the end of the second 20 m/s at the end of the second secondsecond, approximately , approximately 30 m/s at 30 m/s at the end of the third secondthe end of the third second, etc. , etc.

► The velocity of a free-The velocity of a free-falling object which has falling object which has been dropped from a been dropped from a position of rest is position of rest is dependent upon the time dependent upon the time for which it has fallen.for which it has fallen.

►The formula for The formula for determining the determining the velocity of a falling velocity of a falling object after a time of object after a time of t t seconds isseconds is

►

vf = g tvf = g t

► where where gg is the is the acceleration of acceleration of gravity. An gravity. An approximate approximate value for g on value for g on Earth is 10 m/s/s; Earth is 10 m/s/s; more exactly, its more exactly, its value is 9.8 value is 9.8 m/s/s.m/s/s.

Free Fall and the Acceleration of Free Fall and the Acceleration of Gravity Gravity

Example calculations for Example calculations for velocityvelocity►Calculate the Calculate the

velocity of a velocity of a free-falling free-falling object after six, object after six, and eight and eight seconds seconds

►    vf = g tvf = g t

► SolutionSolution► At t = 6 s At t = 6 s ► vf = (10 m/svf = (10 m/s22) (6 s) ) (6 s) ► = 60 m/s= 60 m/s

► At t = 8 sAt t = 8 s► vf = (10 m/svf = (10 m/s22) (8 s) ) (8 s) ► = 80 m/s= 80 m/s

Velocity of FreelyVelocity of Freely FallingFalling BodyBody

► If you were to observe the If you were to observe the motion of a free-falling motion of a free-falling object you would notice that object you would notice that the object averages a the object averages a velocity of velocity of

► 5 m/s in the first second5 m/s in the first second, , ► 15 m/s in the second second15 m/s in the second second, , ► 25 m/s in the third second25 m/s in the third second, , ► 35 m/s in the fourth second35 m/s in the fourth second, ,

etc. etc. ► Our free-falling object would Our free-falling object would

be accelerating at a constant be accelerating at a constant rate.rate.

DistanceDistance►The distance which a free-The distance which a free-

falling object has fallen from falling object has fallen from a position of rest is also a position of rest is also dependent upon the time of dependent upon the time of fall. fall.

►This This distancedistance can be can be computed by use of a computed by use of a formula; the distance fallen formula; the distance fallen after a time of after a time of tt seconds is seconds is given by the formula.given by the formula.

d = 0.5 g td = 0.5 g t22

►where where gg is the acceleration is the acceleration of gravity of gravity

►Calculate the Calculate the distance fallen by distance fallen by a free-falling a free-falling object after one, object after one, two and five two and five seconds seconds

►d = 0.5gtd = 0.5gt22

► Example Calculations:Example Calculations:► At t = 1 s At t = 1 s ► d = (0.5) (10 m/sd = (0.5) (10 m/s22) (1 s)) (1 s)22 ► = 5 m= 5 m

► At t = 2 sAt t = 2 s► d = (0.5) (10 m/sd = (0.5) (10 m/s22) (2 s)) (2 s)22 ► = 20 m= 20 m

► At t = 5 sAt t = 5 s► d = (0.5) (10 m/sd = (0.5) (10 m/s22) (5 s)) (5 s)22 ► = 125 m= 125 m

Example calculations for Example calculations for distancedistance

DistanceDistance► Given these average velocity Given these average velocity

values during each values during each consecutive 1-second time consecutive 1-second time interval, the object falls: interval, the object falls:

► – – 5 meters in the first second5 meters in the first second,,– – 15 meters in the second 15 meters in the second secondsecond (for a total distance of (for a total distance of 20 meters),20 meters),– – 25 meters in the third25 meters in the third secondsecond (for a total distance of (for a total distance of 45 meters),45 meters),– – 35 meters in the fourth 35 meters in the fourth secondsecond (for a total distance of (for a total distance of 80 meters). 80 meters).

Time Time Interval Interval

Average Average Velocity Velocity During Time During Time Interval Interval

Distance Distance Traveled Traveled During Time During Time Interval Interval

Total Distance Total Distance Traveled from Traveled from 0 s to End of 0 s to End of Time Interval Time Interval

0 - 1 s 0 - 1 s 5 m/s 5 m/s 5 m 5 m 5 m 5 m

1 - 2 s 1 - 2 s 15 m/s 15 m/s 15 m 15 m 20 m 20 m

2 - 3 s2 - 3 s 25 m/s 25 m/s 25 m 25 m 45 m 45 m

3 - 4 s 3 - 4 s 35 m/s 35 m/s 35 m 35 m 80 m 80 m

d=0.5gt2v=0.5gt d=dt -di

v=0.5(10m/s2 )1s d= 20 -5

d=0.5(10g/s2 )(1s)2

The table illustrates that a The table illustrates that a free-falling free-falling objectobject which is which is accelerating at a constant accelerating at a constant raterate will will cover different distancescover different distances in in each each

consecutive second. consecutive second. Further analysis of the first and last Further analysis of the first and last

columns of the table above reveal that columns of the table above reveal that there is a there is a square relationship between the square relationship between the

total distance traveled and the time of total distance traveled and the time of traveltravel for an object starting from rest and for an object starting from rest and

moving with a constant acceleration.moving with a constant acceleration.For objects with a constant acceleration, For objects with a constant acceleration,

the the distance of travel is directly distance of travel is directly proportional to the square of the timeproportional to the square of the time of of

travel. travel.

Equation for motion of an Equation for motion of an object in free fall:object in free fall:

gtv

gtvovt

vovg

t

dv

f

tvv

d

gdvv

gttvd

2

2

5.0

0

20

2

20

25.0 gtd

Sample ProblemSample Problem

►A coin was dropped from the A coin was dropped from the top of the LTA building with a top of the LTA building with a height of 727 ft. If there is no height of 727 ft. If there is no air resistance, how fast (ft/s) air resistance, how fast (ft/s) will the coin be moving when it will the coin be moving when it hits the ground?hits the ground?

►215.68 ft/s215.68 ft/s

Solution ASolution A

22

22

22

2

/16

)727(

)/32(5.0727

)/32(5.0

5.0

sft

ftt

tsftft

tsftd

gtd

sftv

ssftv

atv

/68.215

)74.6)(/32( 2

74.6

44.45

t

t

► A marble is dropped from a A marble is dropped from a bridge and strikes the water in bridge and strikes the water in 5 seconds. Calculate the 5 seconds. Calculate the speed with which it strikes and speed with which it strikes and the height of the bridge. the height of the bridge.

► (Vf = 49 m/s, d = 122.5 m) (Vf = 49 m/s, d = 122.5 m)

Problem 1Problem 1

Solution 1Solution 1

smv

ssmv

gtv

f

f

f

/49

)5)(/8.9( 2

md

ssmd

gtd

5.122

)5)(/8.9(5.0

5.022

2

P2P2

► A feather is dropped on the moon A feather is dropped on the moon from a height of 1.40 meters. from a height of 1.40 meters. The acceleration of gravity on The acceleration of gravity on the moon is 1.67 m/sthe moon is 1.67 m/s22. Determine . Determine the time for the feather to fall to the time for the feather to fall to the surface of the moon. the surface of the moon.

► T = 1.29 secT = 1.29 sec

S2S2

?t1.67m/sa

1.40md0m/sv

:Given

2

i

P3P3

►The observation deck of the The observation deck of the World Trade Center is 420 m World Trade Center is 420 m above the street. Determine above the street. Determine the time required for a penny the time required for a penny to free fall from the deck to the to free fall from the deck to the street below. street below.

►T = 9.26 secT = 9.26 sec

S3S3 25.0 gtd

?

/8.9

420

/0

2

t

smg

md

smv

Given

i

sst

ts

tsm

m

tsmm

tsmm

26.97.85

7.85

/9.4

420

)/9.4(420

)/8.9(5.0420

2

22

22

22

22

P4P4

►With what speed in miles/hr With what speed in miles/hr (1 m/s = 2.23 mi/hr) must an (1 m/s = 2.23 mi/hr) must an object be thrown to reach a object be thrown to reach a height of 91.5 m (equivalent to height of 91.5 m (equivalent to one football field)? Assume one football field)? Assume negligible air resistance. negligible air resistance.

►V = 94.4 mi/hrV = 94.4 mi/hr

P4P4

smsmv

smv

smv

msmvsm

gdvv

i

i

i

i

if

/3.42/1793

1793

/17930

)5.91)(/8.9(2/0

2

222

222

222

22

?

?

5.91

/

/8.9

:2

t

v

md

somv

smg

Given

i

f

hrmivsm

hrmismv

i

i

/4.94/1

)/23.2)(/3.42(

P5P5►A 10kg block being held at rest above A 10kg block being held at rest above

the ground is released. The block begins the ground is released. The block begins to fall under only the effect of gravity. At to fall under only the effect of gravity. At the instant that the block is 2.0 meters the instant that the block is 2.0 meters above the ground, the speed of the above the ground, the speed of the block is 2.5m/sec. The block was initially block is 2.5m/sec. The block was initially released at a height of how many released at a height of how many meters. meters.

►D = 2.3 mD = 2.3 m

S5S5

md

mssm

smd

dg

vd

ddg

v

ddgv

v

ddgvv

3.2

2//8.9

)/5.2(5.0

5.0

2

)(2

0

)(2

2

0

2

0

0

2

02

0

020

2

smg

kgm

smv

md

v

ghtinitialheid

/8.9

10

/5.2

2

00

0

AssignmentAssignment► 1. Miguel drops a pile of roof 1. Miguel drops a pile of roof

shingles from the top of a roof shingles from the top of a roof located 8.52 meters above the located 8.52 meters above the ground. Determine the time ground. Determine the time required for the shingles to reach required for the shingles to reach the ground.the ground.

► 2. Brandy throws his mother's 2. Brandy throws his mother's crystal vase vertically upwards crystal vase vertically upwards with an initial velocity of 26.2 with an initial velocity of 26.2 m/s. Determine the height to m/s. Determine the height to which the vase will rise above its which the vase will rise above its initial height.initial height.

► 3. A kangaroo is capable of jumping to a 3. A kangaroo is capable of jumping to a height of 2.62 m. Determine the take-off height of 2.62 m. Determine the take-off speed of the kangaroo.speed of the kangaroo.

► 4. A stone is dropped into a deep well 4. A stone is dropped into a deep well and is heard to hit the water 3.41 s and is heard to hit the water 3.41 s after being dropped. Determine the after being dropped. Determine the depth of the well.depth of the well.

► 5. Ronald McDonald is riding an Air 5. Ronald McDonald is riding an Air Balloon on his way to Subic. If Ronald Balloon on his way to Subic. If Ronald free-falls for 2.6 second, what will be free-falls for 2.6 second, what will be his final velocity and how far will he his final velocity and how far will he fall?fall?

AssignmentAssignment

►Prepare for an EXAM next meeting….Prepare for an EXAM next meeting….

►END OF PRESENTATIONEND OF PRESENTATION

► The ball's average speed for the first 4 seconds The ball's average speed for the first 4 seconds is the average of 0 m/s and 40 m/s, its starting is the average of 0 m/s and 40 m/s, its starting and ending speeds, and distance = average and ending speeds, and distance = average speed times time.speed times time.

► So the object will have fallen 80 meters, and its So the object will have fallen 80 meters, and its speed will be 40 m/s.speed will be 40 m/s.

► A bowling ball falls A bowling ball falls freely (near the freely (near the surface of the surface of the Earth) from rest. Earth) from rest. How far does it fall How far does it fall in 4 seconds, and in 4 seconds, and how fast will it be how fast will it be going at that going at that time?time?

► The position of a The position of a free-falling bodyfree-falling body (neglect (neglect air resistance) under the influence of air resistance) under the influence of gravity can be represented by the function gravity can be represented by the function

where where gg is the acceleration due to gravity is the acceleration due to gravity (on earth ) and s(on earth ) and s00 and v and v00 are the initial are the initial height and velocity of the object (when ). height and velocity of the object (when ).

002

2

1)( stvgtts

► ExampleExample (Falling Object Problem) (Falling Object Problem) A ball is A ball is thrown vertically upward from the ground thrown vertically upward from the ground with an initial velocity of 160 ft/s.with an initial velocity of 160 ft/s.

(a)(a) When will it hit the ground? When will it hit the ground?

        SolutionSolution. Since with and we need . Since with and we need

This is precisely when or and thus the This is precisely when or and thus the ball will hit the ground in 10 seconds.  ball will hit the ground in 10 seconds.  

                          with         and            we need

                                                  

This illustrates that a free-falling object This illustrates that a free-falling object which is accelerating at a constant rate which is accelerating at a constant rate

will cover different distances in each will cover different distances in each consecutive second. consecutive second.

Further analysis of the first and last Further analysis of the first and last columns of the table above reveal that columns of the table above reveal that

there is a square relationship between the there is a square relationship between the total distance traveled and the time of total distance traveled and the time of

travel for an object starting from rest and travel for an object starting from rest and moving with a constant acceleration.moving with a constant acceleration.

For objects with a constant For objects with a constant acceleration, the distance of travel is acceleration, the distance of travel is directly proportional to the square of directly proportional to the square of

the time of travel.the time of travel.

Time Interval Time Interval Average Average Velocity During Velocity During Time Interval Time Interval

Distance Distance Traveled During Traveled During Time Interval Time Interval

Total Distance Total Distance Traveled from 0 Traveled from 0 s to End of Time s to End of Time Interval Interval

0 - 1 s 0 - 1 s 5 m/s 5 m/s 5 m 5 m 5 m 5 m

1 - 2 s 1 - 2 s 15 m/s 15 m/s 15 m 15 m 20 m 20 m

2 - 3 2 - 3 25 m/s 25 m/s 25 m 25 m 45 m 45 m

3 - 4 s 3 - 4 s 35 m/s 35 m/s 35 m 35 m 80 m 80 m

► As such, if an object travels for twice the time, it As such, if an object travels for twice the time, it will cover four times (2will cover four times (222) the distance; the total ) the distance; the total distance traveled after two seconds is four times distance traveled after two seconds is four times the total distance traveled after one second.the total distance traveled after one second.

► If an object travels for three times the time, then it If an object travels for three times the time, then it will cover nine times (3will cover nine times (322) the distance; the distance ) the distance; the distance traveled after three seconds is nine times the traveled after three seconds is nine times the distance traveled after one second.distance traveled after one second.

► Finally, if an object travels for four times the time, Finally, if an object travels for four times the time, then it will cover sixteen times (4then it will cover sixteen times (422) the distance; ) the distance; the distance traveled after four seconds is sixteen the distance traveled after four seconds is sixteen times the distance traveled after one second. times the distance traveled after one second.

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