SlideShare a Scribd company logo
1 of 33
PROJECTILE MOTION
WAYNE FERNANDES
Introduction
 Projectile Motion:
Motion through the air without a propulsion
 Examples:
Part 1.
Motion of Objects Projected
Horizontally
v0
x
y
x
y
x
y
x
y
x
y
x
y
•Motion is accelerated
•Acceleration is constant,
and downward
• a = g = -9.81m/s2
•The horizontal (x)
component of velocity is
constant
•The horizontal and
vertical motions are
independent of each other,
but they have a common
time
g = -9.81m/s2
ANALYSIS OF MOTION
ASSUMPTIONS:
• x-direction (horizontal): uniform motion
• y-direction (vertical): accelerated motion
• no air resistance
QUESTIONS:
• What is the trajectory?
• What is the total time of the motion?
• What is the horizontal range?
• What is the final velocity?
x
y
0
Frame of reference:
h
v0
Equations of motion:
X
Uniform m.
Y
Accel. m.
ACCL. ax = 0 ay = g = -9.81
m/s2
VELC. vx = v0 vy = g t
DSPL. x = v0 t y = h + ½ g t2
g
Trajectory
x = v0 t
y = h + ½ g t2
Eliminate time, t
t = x/v0
y = h + ½ g (x/v0)2
y = h + ½ (g/v0
2
) x2
y = ½ (g/v0
2
) x2
+ h
y
x
h
Parabola, open down
v01
v02 > v01
Total Time, Δt
y = h + ½ g t2
final y = 0 y
x
h
ti =0
tf =Δt
0 = h + ½ g (Δt)2
Solve for Δt:
Δt = √ 2h/(-g)
Δt = √ 2h/(9.81ms-2
)
Total time of motion depends
only on the initial height, h
Δt = tf - ti
Horizontal Range, Δx
final y = 0, time is
the total time Δt
y
x
h
Δt = √ 2h/(-g)
Δx = v0 √ 2h/(-g)
Horizontal range depends on the
initial height, h, and the initial
velocity, v0
Δx
x = v0 t
Δx = v0 Δt
VELOCITY
v
vx = v0
vy = g t
v = √vx
2
+ vy
2
= √v0
2
+g2
t2
tg Θ = v / v = g t / v
Θ
FINAL VELOCITY
v
vx = v0
vy = g t
v = √vx
2
+ vy
2
v = √v0
2
+g2
(2h /(-g))
v = √ v0
2
+ 2h(-g)
Θ
tg Θ = g Δt / v0
= -(-g)√2h/(-g) / v0
= -√2h(-g) / v0
Θ is negative
(below the
horizontal line)
Δt = √ 2h/(-g)
HORIZONTAL THROW - Summary
Trajectory Half -parabola, open
down
Total time Δt = √ 2h/(-g)
Horizontal Range Δx = v0 √ 2h/(-g)
Final Velocity v = √ v0
2
+ 2h(-g)
tg Θ = -√2h(-g) / v0
h – initial height, v0 – initial horizontal velocity, g = -9.81m/s2
Part 2.
Motion of objects projected at an
angle
vi
x
y
θ
vix
viy
Initial velocity: vi = vi [Θ]
Velocity components:
x- direction : vix = vi cos Θ
y- direction : viy = vi sin Θ
Initial position: x = 0, y = 0
x
y
• Motion is accelerated
• Acceleration is constant, and
downward
• a = g = -9.81m/s2
• The horizontal (x) component of
velocity is constant
• The horizontal and vertical
motions are independent of each
other, but they have a common
time
a = g =
- 9.81m/s2
ANALYSIS OF MOTION:
ASSUMPTIONS
• x-direction (horizontal): uniform motion
• y-direction (vertical): accelerated motion
• no air resistance
QUESTIONS
• What is the trajectory?
• What is the total time of the motion?
• What is the horizontal range?
• What is the maximum height?
• What is the final velocity?
Equations of motion:
X
Uniform motion
Y
Accelerated motion
ACCELERATION ax = 0 ay = g = -9.81 m/s2
VELOCITY vx = vix= vi cos Θ
vx = vi cos Θ
vy = viy+ g t
vy = vi sin Θ + g t
DISPLACEMENT x = vix t = vi t cos Θ
x = vi t cos Θ
y = h + viy t + ½ g t2
y = vi tsin Θ + ½ g t2
Equations of motion:
X
Uniform motion
Y
Accelerated motion
ACCELERATION ax = 0 ay = g = -9.81 m/s2
VELOCITY vx = vi cos Θ vy = vi sin Θ + g t
DISPLACEMENT x = vi t cos Θ y = vi tsin Θ + ½ g t2
Trajectory
x = vi t cos Θ
y = vi tsin Θ + ½ g t2
Eliminate time, t
t = x/(vi cos Θ)
y
x
Parabola, open down
2
22
22
2
cos2
tan
cos2cos
sin
x
v
g
xy
v
gx
v
xv
y
i
ii
i
Θ
+Θ=
Θ
+
Θ
Θ
=
y = bx + ax2
Total Time, Δt
final height y = 0, after time interval Δt
0 = vi Δtsin Θ + ½ g (Δt)2
Solve for Δt:
y = vi tsin Θ + ½ g t2
0 = vi sin Θ + ½ g Δt
Δt =
2 vi sin Θ
(-g)
t = 0 Δt
x
Horizontal Range, Δx
final y = 0, time is
the total time Δt
x = vi t cos Θ
Δx = vi Δt cos Θ
x
Δx
y
0
Δt =
2 vi sin Θ
(-g)
Δx =
2vi
2
sin Θ cos Θ
(-g)
Δx =
vi
2
sin (2 Θ)
(-g)
sin (2 Θ) = 2 sin Θ cos Θ
Horizontal Range, Δx
Δx =
vi
2
sin (2 Θ)
(-g)
Θ (deg) sin (2Θ)
0 0.00
15 0.50
30 0.87
45 1.00
60 0.87
75 0.50
90 0
•CONCLUSIONS:
•Horizontal range is greatest for the
throw angle of 450
• Horizontal ranges are the same for
angles Θ and (900
– Θ)
Trajectory and horizontal range
2
22
cos2
tan x
v
g
xy
i Θ
+Θ=
0
5
10
15
20
25
30
35
0 20 40 60 80
15 deg
30 deg
45 deg
60 deg
75 deg
vi = 25 m/s
Velocity
•Final speed = initial speed (conservation of energy)
•Impact angle = - launch angle (symmetry of parabola)
Maximum Height
vy = vi sin Θ + g t
y = vi tsin Θ + ½ g t2
At maximum height vy = 0
0 = vi sin Θ + g tup
tup =
vi sin Θ
(-g)
tup = Δt/2
hmax = vi tupsin Θ + ½ g tup
2
hmax = vi
2
sin2
Θ/(-g) + ½ g(vi
2
sin2
Θ)/g2
hmax =
vi
2
sin2
Θ
2(-g)
Projectile Motion – Final Equations
Trajectory Parabola, open down
Total time Δt =
Horizontal range Δx =
Max height hmax =
(0,0) – initial position, vi = vi [Θ]– initial velocity, g = -9.81m/s2
2 vi sin Θ
(-g)
vi
2
sin (2 Θ)
(-g)
vi
2
sin2
Θ
2(-g)
PROJECTILE MOTION - SUMMARY
 Projectile motion is motion with a constant
horizontal velocity combined with a constant
vertical acceleration
 The projectile moves along a parabola
THANK YOU

More Related Content

What's hot

Fluid dynamics 1
Fluid dynamics 1Fluid dynamics 1
Fluid dynamics 1guest7b51c7
 
Inertial frame of reference
Inertial frame of referenceInertial frame of reference
Inertial frame of referenceJAGADEESH S S
 
Projectile motion 2
Projectile motion 2Projectile motion 2
Projectile motion 2mattvijay
 
PAp physics 1&2_-_projectile_motion
PAp physics 1&2_-_projectile_motionPAp physics 1&2_-_projectile_motion
PAp physics 1&2_-_projectile_motionkampkorten
 
04-14-08 - Momentum And Impulse
04-14-08 - Momentum And Impulse04-14-08 - Momentum And Impulse
04-14-08 - Momentum And Impulsewjerlinger
 
Kinematic equations of motion
Kinematic equations of motionKinematic equations of motion
Kinematic equations of motionmantlfin
 
Uniformly accelerated motion
Uniformly accelerated motionUniformly accelerated motion
Uniformly accelerated motionPRINTDESK by Dan
 
S3 Chapter 1 Introduction of Fluid
S3 Chapter 1 Introduction of FluidS3 Chapter 1 Introduction of Fluid
S3 Chapter 1 Introduction of Fluidno suhaila
 
5.8 rectilinear motion
5.8 rectilinear motion5.8 rectilinear motion
5.8 rectilinear motiondicosmo178
 
Waves ppt.
Waves ppt.Waves ppt.
Waves ppt.hlayala
 
Amplitude and loudness
Amplitude and loudnessAmplitude and loudness
Amplitude and loudnessAdam Sanders
 

What's hot (20)

Fluid dynamics 1
Fluid dynamics 1Fluid dynamics 1
Fluid dynamics 1
 
Inertial frame of reference
Inertial frame of referenceInertial frame of reference
Inertial frame of reference
 
Properties of the fluids
Properties of the fluidsProperties of the fluids
Properties of the fluids
 
Continuity Equation
Continuity EquationContinuity Equation
Continuity Equation
 
Projectile motion 2
Projectile motion 2Projectile motion 2
Projectile motion 2
 
PAp physics 1&2_-_projectile_motion
PAp physics 1&2_-_projectile_motionPAp physics 1&2_-_projectile_motion
PAp physics 1&2_-_projectile_motion
 
Motion in one dimension
Motion in one dimensionMotion in one dimension
Motion in one dimension
 
04-14-08 - Momentum And Impulse
04-14-08 - Momentum And Impulse04-14-08 - Momentum And Impulse
04-14-08 - Momentum And Impulse
 
Projectile motion
Projectile motionProjectile motion
Projectile motion
 
Fil. report
Fil. reportFil. report
Fil. report
 
Kinematic equations of motion
Kinematic equations of motionKinematic equations of motion
Kinematic equations of motion
 
Circular motion
Circular motion Circular motion
Circular motion
 
Waves
WavesWaves
Waves
 
Uniformly accelerated motion
Uniformly accelerated motionUniformly accelerated motion
Uniformly accelerated motion
 
S3 Chapter 1 Introduction of Fluid
S3 Chapter 1 Introduction of FluidS3 Chapter 1 Introduction of Fluid
S3 Chapter 1 Introduction of Fluid
 
Waves
WavesWaves
Waves
 
5.8 rectilinear motion
5.8 rectilinear motion5.8 rectilinear motion
5.8 rectilinear motion
 
Waves ppt.
Waves ppt.Waves ppt.
Waves ppt.
 
Amplitude and loudness
Amplitude and loudnessAmplitude and loudness
Amplitude and loudness
 
Free fall
Free fallFree fall
Free fall
 

Similar to Projectile motion

Projectile motion
Projectile motionProjectile motion
Projectile motionTekZeno
 
03 Motion in Two & Three Dimensions.ppt
03 Motion in Two & Three Dimensions.ppt03 Motion in Two & Three Dimensions.ppt
03 Motion in Two & Three Dimensions.pptCharleneMaeADotillos
 
Projectile motionchemistory (4)
Projectile motionchemistory (4)Projectile motionchemistory (4)
Projectile motionchemistory (4)Sahil Raturi
 
4. Motion in a Plane 1.pptx.pdf
4. Motion in a Plane 1.pptx.pdf4. Motion in a Plane 1.pptx.pdf
4. Motion in a Plane 1.pptx.pdfMKumarVarnana
 
PROJECTILE MOTION-Horizontal and Vertical
PROJECTILE MOTION-Horizontal and VerticalPROJECTILE MOTION-Horizontal and Vertical
PROJECTILE MOTION-Horizontal and VerticalMAESTRELLAMesa2
 
Ch 12 (4) Curvilinear Motion X-Y Coordinate.pptx
Ch 12 (4) Curvilinear Motion X-Y  Coordinate.pptxCh 12 (4) Curvilinear Motion X-Y  Coordinate.pptx
Ch 12 (4) Curvilinear Motion X-Y Coordinate.pptxBilalHassan124013
 
projectile motion grade 9-170213175803.pptx
projectile motion grade 9-170213175803.pptxprojectile motion grade 9-170213175803.pptx
projectile motion grade 9-170213175803.pptxPrincessRegunton
 
2 2-d kinematics notes
2   2-d kinematics notes2   2-d kinematics notes
2 2-d kinematics notescpphysics
 
Projectile motion
Projectile motionProjectile motion
Projectile motiongngr0810
 
Projectile motion
Projectile motionProjectile motion
Projectile motiongngr0810
 
Motion Graph & equations
Motion Graph & equationsMotion Graph & equations
Motion Graph & equationsNurul Fadhilah
 
projectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdfprojectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdfMaAnnFuriscal3
 
projectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdfprojectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdfMaAnnFuriscal3
 
Lecture curvilinear
Lecture curvilinearLecture curvilinear
Lecture curvilinearMustafa Bzu
 

Similar to Projectile motion (20)

Projectile motion
Projectile motionProjectile motion
Projectile motion
 
projectile motion.ppt
projectile motion.pptprojectile motion.ppt
projectile motion.ppt
 
Projectile motion
Projectile motionProjectile motion
Projectile motion
 
Projectile motion (2)
Projectile motion (2)Projectile motion (2)
Projectile motion (2)
 
Ch#4 MOTION IN 2 DIMENSIONS
Ch#4 MOTION IN 2 DIMENSIONSCh#4 MOTION IN 2 DIMENSIONS
Ch#4 MOTION IN 2 DIMENSIONS
 
03 Motion in Two & Three Dimensions.ppt
03 Motion in Two & Three Dimensions.ppt03 Motion in Two & Three Dimensions.ppt
03 Motion in Two & Three Dimensions.ppt
 
Projectile motionchemistory (4)
Projectile motionchemistory (4)Projectile motionchemistory (4)
Projectile motionchemistory (4)
 
Chapter 3
Chapter 3Chapter 3
Chapter 3
 
4. Motion in a Plane 1.pptx.pdf
4. Motion in a Plane 1.pptx.pdf4. Motion in a Plane 1.pptx.pdf
4. Motion in a Plane 1.pptx.pdf
 
PROJECTILE MOTION-Horizontal and Vertical
PROJECTILE MOTION-Horizontal and VerticalPROJECTILE MOTION-Horizontal and Vertical
PROJECTILE MOTION-Horizontal and Vertical
 
Ch 12 (4) Curvilinear Motion X-Y Coordinate.pptx
Ch 12 (4) Curvilinear Motion X-Y  Coordinate.pptxCh 12 (4) Curvilinear Motion X-Y  Coordinate.pptx
Ch 12 (4) Curvilinear Motion X-Y Coordinate.pptx
 
projectile motion grade 9-170213175803.pptx
projectile motion grade 9-170213175803.pptxprojectile motion grade 9-170213175803.pptx
projectile motion grade 9-170213175803.pptx
 
2 2-d kinematics notes
2   2-d kinematics notes2   2-d kinematics notes
2 2-d kinematics notes
 
PROJECTILE MOTION
PROJECTILE MOTIONPROJECTILE MOTION
PROJECTILE MOTION
 
Projectile motion
Projectile motionProjectile motion
Projectile motion
 
Projectile motion
Projectile motionProjectile motion
Projectile motion
 
Motion Graph & equations
Motion Graph & equationsMotion Graph & equations
Motion Graph & equations
 
projectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdfprojectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdf
 
projectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdfprojectile motion horizontal vertical -170213175803 (1).pdf
projectile motion horizontal vertical -170213175803 (1).pdf
 
Lecture curvilinear
Lecture curvilinearLecture curvilinear
Lecture curvilinear
 

More from WAYNE FERNANDES (20)

Physics Formula list (4)
Physics Formula list (4)Physics Formula list (4)
Physics Formula list (4)
 
Physics Formula list (3)
Physics Formula list (3)Physics Formula list (3)
Physics Formula list (3)
 
Physics Formula list (1)
Physics Formula list (1)Physics Formula list (1)
Physics Formula list (1)
 
Refraction of light
Refraction of lightRefraction of light
Refraction of light
 
electric fields forces
electric fields forceselectric fields forces
electric fields forces
 
ray optics
 ray optics ray optics
ray optics
 
Fluids
 Fluids Fluids
Fluids
 
energy work
 energy work energy work
energy work
 
linear momentum
 linear momentum linear momentum
linear momentum
 
SOUND WAVES
SOUND WAVESSOUND WAVES
SOUND WAVES
 
Scalars and vectors
Scalars and vectorsScalars and vectors
Scalars and vectors
 
Chromatography
Chromatography Chromatography
Chromatography
 
STRUCTURE OF ATOM
STRUCTURE OF ATOMSTRUCTURE OF ATOM
STRUCTURE OF ATOM
 
Alkane alkene alkyne
Alkane alkene alkyneAlkane alkene alkyne
Alkane alkene alkyne
 
FORMULA LIST FOR PHYSICS
FORMULA LIST FOR PHYSICSFORMULA LIST FOR PHYSICS
FORMULA LIST FOR PHYSICS
 
HUMAN SKELETON AND LOCOMOTION
HUMAN SKELETON AND LOCOMOTIONHUMAN SKELETON AND LOCOMOTION
HUMAN SKELETON AND LOCOMOTION
 
HUMAN NUTRITION
HUMAN NUTRITIONHUMAN NUTRITION
HUMAN NUTRITION
 
STUDY OF ANIMAL TISSUES
STUDY OF ANIMAL TISSUESSTUDY OF ANIMAL TISSUES
STUDY OF ANIMAL TISSUES
 
Kingdom Animalia
Kingdom AnimaliaKingdom Animalia
Kingdom Animalia
 
Mitosis and meiosis
Mitosis and meiosisMitosis and meiosis
Mitosis and meiosis
 

Recently uploaded

The Influence and Evolution of Mogul Press in Contemporary Public Relations.docx
The Influence and Evolution of Mogul Press in Contemporary Public Relations.docxThe Influence and Evolution of Mogul Press in Contemporary Public Relations.docx
The Influence and Evolution of Mogul Press in Contemporary Public Relations.docxMogul Press
 
Microsoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdf
Microsoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdfMicrosoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdf
Microsoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdfSkillCertProExams
 
Databricks Machine Learning Associate Exam Dumps 2024.pdf
Databricks Machine Learning Associate Exam Dumps 2024.pdfDatabricks Machine Learning Associate Exam Dumps 2024.pdf
Databricks Machine Learning Associate Exam Dumps 2024.pdfSkillCertProExams
 
Breathing in New Life_ Part 3 05 22 2024.pptx
Breathing in New Life_ Part 3 05 22 2024.pptxBreathing in New Life_ Part 3 05 22 2024.pptx
Breathing in New Life_ Part 3 05 22 2024.pptxFamilyWorshipCenterD
 
ServiceNow CIS-Discovery Exam Dumps 2024
ServiceNow CIS-Discovery Exam Dumps 2024ServiceNow CIS-Discovery Exam Dumps 2024
ServiceNow CIS-Discovery Exam Dumps 2024SkillCertProExams
 
ACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdf
ACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdfACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdf
ACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdfKinben Innovation Private Limited
 
Deciding The Topic of our Magazine.pptx.
Deciding The Topic of our Magazine.pptx.Deciding The Topic of our Magazine.pptx.
Deciding The Topic of our Magazine.pptx.bazilnaeem7
 
Understanding Poverty: A Community Questionnaire
Understanding Poverty: A Community QuestionnaireUnderstanding Poverty: A Community Questionnaire
Understanding Poverty: A Community Questionnairebazilnaeem7
 
2024-05-15-Surat Meetup-Hyperautomation.pptx
2024-05-15-Surat Meetup-Hyperautomation.pptx2024-05-15-Surat Meetup-Hyperautomation.pptx
2024-05-15-Surat Meetup-Hyperautomation.pptxnitishjain2015
 
DAY 0 8 A Revelation 05-19-2024 PPT.pptx
DAY 0 8 A Revelation 05-19-2024 PPT.pptxDAY 0 8 A Revelation 05-19-2024 PPT.pptx
DAY 0 8 A Revelation 05-19-2024 PPT.pptxFamilyWorshipCenterD
 

Recently uploaded (10)

The Influence and Evolution of Mogul Press in Contemporary Public Relations.docx
The Influence and Evolution of Mogul Press in Contemporary Public Relations.docxThe Influence and Evolution of Mogul Press in Contemporary Public Relations.docx
The Influence and Evolution of Mogul Press in Contemporary Public Relations.docx
 
Microsoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdf
Microsoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdfMicrosoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdf
Microsoft Fabric Analytics Engineer (DP-600) Exam Dumps 2024.pdf
 
Databricks Machine Learning Associate Exam Dumps 2024.pdf
Databricks Machine Learning Associate Exam Dumps 2024.pdfDatabricks Machine Learning Associate Exam Dumps 2024.pdf
Databricks Machine Learning Associate Exam Dumps 2024.pdf
 
Breathing in New Life_ Part 3 05 22 2024.pptx
Breathing in New Life_ Part 3 05 22 2024.pptxBreathing in New Life_ Part 3 05 22 2024.pptx
Breathing in New Life_ Part 3 05 22 2024.pptx
 
ServiceNow CIS-Discovery Exam Dumps 2024
ServiceNow CIS-Discovery Exam Dumps 2024ServiceNow CIS-Discovery Exam Dumps 2024
ServiceNow CIS-Discovery Exam Dumps 2024
 
ACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdf
ACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdfACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdf
ACM CHT Best Inspection Practices Kinben Innovation MIC Slideshare.pdf
 
Deciding The Topic of our Magazine.pptx.
Deciding The Topic of our Magazine.pptx.Deciding The Topic of our Magazine.pptx.
Deciding The Topic of our Magazine.pptx.
 
Understanding Poverty: A Community Questionnaire
Understanding Poverty: A Community QuestionnaireUnderstanding Poverty: A Community Questionnaire
Understanding Poverty: A Community Questionnaire
 
2024-05-15-Surat Meetup-Hyperautomation.pptx
2024-05-15-Surat Meetup-Hyperautomation.pptx2024-05-15-Surat Meetup-Hyperautomation.pptx
2024-05-15-Surat Meetup-Hyperautomation.pptx
 
DAY 0 8 A Revelation 05-19-2024 PPT.pptx
DAY 0 8 A Revelation 05-19-2024 PPT.pptxDAY 0 8 A Revelation 05-19-2024 PPT.pptx
DAY 0 8 A Revelation 05-19-2024 PPT.pptx
 

Projectile motion

  • 2. Introduction  Projectile Motion: Motion through the air without a propulsion  Examples:
  • 3. Part 1. Motion of Objects Projected Horizontally
  • 5. x y
  • 6. x y
  • 7. x y
  • 8. x y
  • 9. x y •Motion is accelerated •Acceleration is constant, and downward • a = g = -9.81m/s2 •The horizontal (x) component of velocity is constant •The horizontal and vertical motions are independent of each other, but they have a common time g = -9.81m/s2
  • 10. ANALYSIS OF MOTION ASSUMPTIONS: • x-direction (horizontal): uniform motion • y-direction (vertical): accelerated motion • no air resistance QUESTIONS: • What is the trajectory? • What is the total time of the motion? • What is the horizontal range? • What is the final velocity?
  • 11. x y 0 Frame of reference: h v0 Equations of motion: X Uniform m. Y Accel. m. ACCL. ax = 0 ay = g = -9.81 m/s2 VELC. vx = v0 vy = g t DSPL. x = v0 t y = h + ½ g t2 g
  • 12. Trajectory x = v0 t y = h + ½ g t2 Eliminate time, t t = x/v0 y = h + ½ g (x/v0)2 y = h + ½ (g/v0 2 ) x2 y = ½ (g/v0 2 ) x2 + h y x h Parabola, open down v01 v02 > v01
  • 13. Total Time, Δt y = h + ½ g t2 final y = 0 y x h ti =0 tf =Δt 0 = h + ½ g (Δt)2 Solve for Δt: Δt = √ 2h/(-g) Δt = √ 2h/(9.81ms-2 ) Total time of motion depends only on the initial height, h Δt = tf - ti
  • 14. Horizontal Range, Δx final y = 0, time is the total time Δt y x h Δt = √ 2h/(-g) Δx = v0 √ 2h/(-g) Horizontal range depends on the initial height, h, and the initial velocity, v0 Δx x = v0 t Δx = v0 Δt
  • 15. VELOCITY v vx = v0 vy = g t v = √vx 2 + vy 2 = √v0 2 +g2 t2 tg Θ = v / v = g t / v Θ
  • 16. FINAL VELOCITY v vx = v0 vy = g t v = √vx 2 + vy 2 v = √v0 2 +g2 (2h /(-g)) v = √ v0 2 + 2h(-g) Θ tg Θ = g Δt / v0 = -(-g)√2h/(-g) / v0 = -√2h(-g) / v0 Θ is negative (below the horizontal line) Δt = √ 2h/(-g)
  • 17. HORIZONTAL THROW - Summary Trajectory Half -parabola, open down Total time Δt = √ 2h/(-g) Horizontal Range Δx = v0 √ 2h/(-g) Final Velocity v = √ v0 2 + 2h(-g) tg Θ = -√2h(-g) / v0 h – initial height, v0 – initial horizontal velocity, g = -9.81m/s2
  • 18. Part 2. Motion of objects projected at an angle
  • 19. vi x y θ vix viy Initial velocity: vi = vi [Θ] Velocity components: x- direction : vix = vi cos Θ y- direction : viy = vi sin Θ Initial position: x = 0, y = 0
  • 20. x y • Motion is accelerated • Acceleration is constant, and downward • a = g = -9.81m/s2 • The horizontal (x) component of velocity is constant • The horizontal and vertical motions are independent of each other, but they have a common time a = g = - 9.81m/s2
  • 21. ANALYSIS OF MOTION: ASSUMPTIONS • x-direction (horizontal): uniform motion • y-direction (vertical): accelerated motion • no air resistance QUESTIONS • What is the trajectory? • What is the total time of the motion? • What is the horizontal range? • What is the maximum height? • What is the final velocity?
  • 22. Equations of motion: X Uniform motion Y Accelerated motion ACCELERATION ax = 0 ay = g = -9.81 m/s2 VELOCITY vx = vix= vi cos Θ vx = vi cos Θ vy = viy+ g t vy = vi sin Θ + g t DISPLACEMENT x = vix t = vi t cos Θ x = vi t cos Θ y = h + viy t + ½ g t2 y = vi tsin Θ + ½ g t2
  • 23. Equations of motion: X Uniform motion Y Accelerated motion ACCELERATION ax = 0 ay = g = -9.81 m/s2 VELOCITY vx = vi cos Θ vy = vi sin Θ + g t DISPLACEMENT x = vi t cos Θ y = vi tsin Θ + ½ g t2
  • 24. Trajectory x = vi t cos Θ y = vi tsin Θ + ½ g t2 Eliminate time, t t = x/(vi cos Θ) y x Parabola, open down 2 22 22 2 cos2 tan cos2cos sin x v g xy v gx v xv y i ii i Θ +Θ= Θ + Θ Θ = y = bx + ax2
  • 25. Total Time, Δt final height y = 0, after time interval Δt 0 = vi Δtsin Θ + ½ g (Δt)2 Solve for Δt: y = vi tsin Θ + ½ g t2 0 = vi sin Θ + ½ g Δt Δt = 2 vi sin Θ (-g) t = 0 Δt x
  • 26. Horizontal Range, Δx final y = 0, time is the total time Δt x = vi t cos Θ Δx = vi Δt cos Θ x Δx y 0 Δt = 2 vi sin Θ (-g) Δx = 2vi 2 sin Θ cos Θ (-g) Δx = vi 2 sin (2 Θ) (-g) sin (2 Θ) = 2 sin Θ cos Θ
  • 27. Horizontal Range, Δx Δx = vi 2 sin (2 Θ) (-g) Θ (deg) sin (2Θ) 0 0.00 15 0.50 30 0.87 45 1.00 60 0.87 75 0.50 90 0 •CONCLUSIONS: •Horizontal range is greatest for the throw angle of 450 • Horizontal ranges are the same for angles Θ and (900 – Θ)
  • 28. Trajectory and horizontal range 2 22 cos2 tan x v g xy i Θ +Θ= 0 5 10 15 20 25 30 35 0 20 40 60 80 15 deg 30 deg 45 deg 60 deg 75 deg vi = 25 m/s
  • 29. Velocity •Final speed = initial speed (conservation of energy) •Impact angle = - launch angle (symmetry of parabola)
  • 30. Maximum Height vy = vi sin Θ + g t y = vi tsin Θ + ½ g t2 At maximum height vy = 0 0 = vi sin Θ + g tup tup = vi sin Θ (-g) tup = Δt/2 hmax = vi tupsin Θ + ½ g tup 2 hmax = vi 2 sin2 Θ/(-g) + ½ g(vi 2 sin2 Θ)/g2 hmax = vi 2 sin2 Θ 2(-g)
  • 31. Projectile Motion – Final Equations Trajectory Parabola, open down Total time Δt = Horizontal range Δx = Max height hmax = (0,0) – initial position, vi = vi [Θ]– initial velocity, g = -9.81m/s2 2 vi sin Θ (-g) vi 2 sin (2 Θ) (-g) vi 2 sin2 Θ 2(-g)
  • 32. PROJECTILE MOTION - SUMMARY  Projectile motion is motion with a constant horizontal velocity combined with a constant vertical acceleration  The projectile moves along a parabola