PROJECTILE MOTION

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:

y = h + ½ g t2x = v0 tDSPL.

vy = g tvx = v0VELC.

ay = g = -9.81 m/s2

ax = 0ACCL.

Y

Accel. m.

X

Uniform m.

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/v02) x2

y = ½ (g/v02) x2 + h

y

x

hParabola, open down

v01v02 > v01

Total Time, Δty = h + ½ g t2

final y = 0 y

x

hti =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 tv = √vx

2 + vy2

= √v02+g2t2

tg Θ = vy

/ vx

= g t / v0

Θ

FINAL VELOCITY

v

vx = v0

vy = g tv = √vx

2 + vy2

v = √v02+g2(2h /(-g))

v = √ v02+ 2h(-g)

Θtg Θ = g Δt / v0

= -(-g)√2h/(-g) / v0

= -√2h(-g) / v0

Θ is negative (below the horizontal line)

Δt = √ 2h/(-g)

HORIZONTAL THROW - Summary

v = √ v02+ 2h(-g)

tg Θ = -√2h(-g) / v0

Final Velocity

Δx = v0 √ 2h/(-g)Horizontal Range

Δt = √ 2h/(-g)Total time

Half -parabola, open down

Trajectory

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 : v0x = v0 cos Θ

y- direction : v0y = v0 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:

y = v0 t sin Θ + ½ g t2

x = v0x t = v0 t cos Θ

x = v0 t cos Θ

DISPLACEMENT

vy = v0 sin Θ - g t

vx = v0x= v0 cos ΘVELOCITY

ay = g = -9.81 m/s2ax = 0ACCELERATION

Y

Accelerated motion

X

Uniform motion

Trajectoryx = vi t cos Θ

y = vi t sin Θ + ½ g t2

Eliminate time, t

t = x/(vi cos Θ)

y

x

Parabola, open down

222

22

2

cos2tan

cos2cos

sin

xv

gxy

v

gx

v

xvy

i

ii

i

Θ+Θ=

Θ+

ΘΘ=

y = bx + ax2

Total Time, Δt

final height y = 0, after time interval Δt

0 = v0 Δt sin Θ + ½ g (Δt)2

Solve for Δt:

y = v0 t sin Θ + ½ g t2

0 = v0 sin Θ + ½ g Δt

Δt = 2 v0 sin Θ

(-g)t = 0 Δt

x

Horizontal Range, Δx

final y = 0, time is the total time Δt

x = v0 t cos Θ

Δx = vo Δt cos Θ

x

Δx

y

0

Δt =2 v0sin Θ

(-g)

Δx =2vo

2 sin Θ cos Θ

(-g)Δx =

vo 2 sin (2 Θ)(-g)

sin (2 Θ) = 2 sin Θ cos Θ

Horizontal Range, Δx

Δx =v0

2 sin (2 Θ)(-g)

0.5075

0.000

090

0.8760

1.0045

0.8730

0.5015

sin (2 Θ)Θ (deg) •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

222 cos2

tan xv

gxy

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 = vosin Θ + g t

y = v0 t sin Θ + ½ g t2

At maximum height vy = 0

0 = vi0sin Θ + g tup

tup = v0 sin Θ

(-g)

tup = Δt/2

hmax = v0 t upsin Θ + ½ g tup2

hmax = v02

sin2 Θ/(-g) + ½ g(vi2 sin2 Θ)/g2

hmax =

vi2

sin2 Θ

2(-g)

Projectile Motion – Final Equations

hmax =Max height

Δx =Horizontal range

Δt =Total time

Parabola, open downTrajectory

(0,0) – initial position, vi = vi [Θ]– initial velocity, g = -9.81m/s2

2 v0sin Θ

(-g)

v02 sin (2 Θ)

(-g)

v02 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