projectile motion

projectile motion - Transcript

PROJECTILE MOTION Senior High School Physics
Lech Jedral 2006
Part 1 Part 2

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Introduction
Projectile

Motion Motion through the air without a propulsion Examples

Part 1 Motion of Objects Projected Horizontally

y

v0

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 x common time

g 9 81m s2

ANALYSIS OF MOTION
ASSUMPTIONS x direction horizontal y direction vertical no air resistance uniform motion accelerated motion

QUESTIONS What is the trajectory What is the total time of the motion What is the horizontal range What is the final velocity

Frame of reference y v0

Equations of motion

X Uniform m ACCL ax 0 vx v0 x v0 t

Y Accel m ay g 9 81 m s2 vy g t y h g t2

h

g

VELC
0 x DSPL

Trajectory
x v0 t y h g t2
Eliminate time t t x v0 y h g x v0 2 h v02 v01 y Parabola open down

v01

y h g v02 x2 y g v02 x2 h
x

Total Time t
y h g t2
final y 0 0 h g t 2 Solve for t h y ti 0

t tf ti

t 2h g t 2h 9 81ms 2
Total time of motion depends only on the initial height h tf t x

Horizontal Range x
x v0 t
final y 0 time is the total time t y h

x v 0 t t 2h g x v0 2h g
Horizontal range depends on the initial height h and the initial velocity v0

x
x

VELOCITY
vx v0 vy g t


v v

2 x

vy

2

v

v02 g2t2

tg v v g t v

FINAL VELOCITY
vx v0

t 2h g
v v
2 x

vy

2

vy g t



tg g t v0

v

g 2h g v0

v v02 g2 2h g v v02 2h g

2h g v0
is negative below the horizontal line

HORIZONTAL THROW Summary
h initial height v0 initial horizontal velocity g 9 81m s2

Trajectory Total time Horizontal Range Final Velocity

Half parabola open down

t 2h g x v0 2h g v v02 2h g tg 2h g v0

Part 2 Motion of objects projected at an angle

y

vi

Initial position x 0 y 0 Initial velocity vi vi

viy Velocity components x direction vix vi cos vix y direction viy vi sin x

y a g 9 81m s2
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

x

ANALYSIS OF MOTION
ASSUMPTIONS x direction horizontal y direction vertical no air resistance uniform motion accelerated motion

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
ACCELERATION

Y Accelerated motion ay g 9 81 m s2 vy viy g t vy vi sin g t y h viy t g t2 y vi t sin g t2

ax 0 vx vix vi cos vx vi cos x vix t vi t cos x vi t cos

VELOCITY

DISPLACEMENT

Equations of motion X Uniform motion
ACCELERATION

Y Accelerated motion ay g 9 81 m s2 vy vi sin g t y vi t sin g t2

ax 0 vx vi cos x vi t cos

VELOCITY

DISPLACEMENT

x vi t cos
Eliminate time t

Trajectory
y Parabola open down

y vi t sin g t2
t x vi cos

vi x sin gx 2 y 2 vi cos 2vi cos 2 y x tan g x2 2vi2 cos 2

y bx ax2 x

Total Time t
y vi t sin g t2
final height y 0 after time interval t 0 vi t sin g t 2 Solve for t 0 vi sin g t 2 vi sin g x

t

t 0

t

Horizontal Range x
x vi t cos
final y 0 time is the total time t y

x vi t cos t
2 vi sin g
sin 2 2 sin cos

0

x

x vi 2 sin 2
g

x

2vi 2 sin cos
g

x

Horizontal Range x
x
deg sin 2
0 15 30 45 60 75 90 0 00 0 50 0 87 1 00 0 87 0 50 0

vi 2 sin 2
g CONCLUSIONS Horizontal range is greatest for the throw angle of 450

Horizontal ranges are the same for angles and 900

Trajectory and horizontal range
g y x tan 2 x2 2vi cos 2
35 30 25 20 15 10 5 0 0 20 40 60 80

vi 25 m s

15 deg 30 deg 45 deg 60 deg 75 deg

Velocity

Final speed initial speed conservation of energy Impact angle launch angle symmetry of parabola

Maximum Height
vy vi sin g t y vi t sin g t2
At maximum height vy 0

0 vi sin g tup tup vi sin
g

hmax vi t upsin g tup2 hmax vi2 sin2 g g vi2 sin2 g2 vi2 sin2 hmax 2 g

tup t 2

Projectile Motion Final Equations
0 0 initial position vi vi initial velocity g 9 81m s2 Trajectory
Parabola open down

Total time

t

2 vi sin g vi 2 sin 2 g

Horizontal range

x

Max height

hmax

vi2 sin2 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