Bahan Kuliah: Gerak Satu Dan Dua Dimensi

8/19/2019 Bahan Kuliah: Gerak Satu Dan Dua Dimensi

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Motion in one and two dimention

Setyawan P. Sakti

FISIKA I


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Position & Displacement


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Dynamics

• The branch of physics involving the motion of anobject and the relationship between that motionand other physics concepts

• K inemat ics is a part of dynamics – In kinematics, you are interested in the description of

motion

– Not concerned with the cause of the motion


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Position

• Position is defined in

terms of a frame of

reference

Frame A: x i >0 and x f >0Frame B: x’ i 0

• One dimensional, so

generally the x- or y-axis

A

B

x’ xi’ x f ’

Units

Feet (ft)US Cust

Centimeters (cm)CGS

Meters (m)SI

Units

Feet (ft)US Cust

Centimeters (cm)CGS

Meters (m)SI


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Displacement

• Displacement measures thechange in position

– Represented as x (ifhorizontal) or y (if vertical)

– Vector quantity (i.e. needs

directional information)• + or - is generally sufficient to

indicate direction for one-dimensional motion

UnitsSI Meters (m)

CGS Centimeters (cm)

US Cust Feet (ft)


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Displacement

m

mm

x x x i f

70

1080

1

m

mm

x x x i f

60

8020

2

• Displacement measures the change in position

– Represented as x (if horizontal) or y (if vertical)

– Vector quantity (i.e. needs directional information)• + or - is generally sufficient to indicate direction for one-dimensional motion


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Distance or Displacement?

a x

• Distance may be, but is not necessarily, the magnitude ofthe displacement

Distance(blue line)

Displacement

(red line)

A B

b x x


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Speed and Position


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Velocity


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Speed

• Speed is a scalar quantity (no information about

sign/direction is need)

– same units as velocity

– Average speed = total distance / total time

• Speed is the magnitude of the velocity


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The velocity vector

• The velocity of an objecttells you both its speed and

its direction of motion.• A velocity can be positive or

negative.

• The positive or negative

sign for velocity is based onthe calculation of a changein position.

Two cars going opposite

directions have the same

speed, but their

velocities are different—

one is positive and the

other is negative.


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The velocity vector

• Velocity is the change in position divided by the

change in time.


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Average Velocity

• It takes time for an object to undergo a displacement

• The average velocity is rate at which the displacement

occurs

• Direction will be the same as the direction of the

displacement (t is always positive)

t

x x

t

x

v i f

average


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Units of Velocity

Units

SI Meters per second (m/s)

CGS Centimeters per second (cm/s)

US Customary Feet per second (ft/s)


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Example:

Suppose that in both cases truck

covers the distance in 10 seconds:

sm

s

m

t

x

v average

7

10

701

1

sm

s

m

t

xv average

6

10

602

2


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• Velocity can be determined from a position-timegraph

• Average velocity equals the slope of the line joiningthe initial and final positions

Graphical Interpretation of Average Velocity

sm

s

m

t

x

vaverage

13

0.3

40


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Instantaneous Velocity

• Instantaneous velocity is defined as the limit of the average

velocity as the time interval becomes infinitesimally short,

or as the time interval approaches zero

• The instantaneous velocity indicates what is happening at

every point of time

dt

xd x x xv

i f

t t inst

ΔtlimΔtlim 00

Δ


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Uniform Velocity

• Uniform velocity is constant velocity

• The instantaneous velocities are always the same

– All the instantaneous velocities will also equal the

average velocity


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Graphical Interpretation of Instantaneous Velocity

• Instantaneous velocity is the slope of the tangent to the curve at

the time of interest

• The instantaneous speed is the magnitude of the instantaneous

velocity


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Interpreting a distance versus

time graph

1. How many stops does itmake?

2. What is the boat’s averagespeed for the whole trip?

3. What is the highest speed theboat reaches?

This distance versus time graph shows a boat travelingthrough a long canal. The boat has to stop at locks for

changes in water level.


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Average vs Instantaneous Velocity

Average velocity Instantaneous velocity


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Average Acceleration

• Changing velocity (non-uniform) means an

acceleration is present

• Average acceleration is the rate of change of the

velocity

• Average acceleration is a vector quantity (i.e.described by both magnitude and direction)

t

vv

t

va

i f

average


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I t t d U if


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Instantaneous and Uniform

Acceleration

• Instantaneous acceleration is the limit of the averageacceleration as the time interval goes to zero

• When the instantaneous accelerations are always the same,the acceleration will be uniform

– The instantaneous accelerations will all be equal to theaverage acceleration

0 0

lim lim f i

inst t t

v vva

t t

dt

dv


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Graphical Interpretation of

Acceleration

• Average acceleration is the

slope of the line connecting the

initial and final velocities on a

velocity-time graph

• Instantaneous acceleration is the

slope of the tangent to the curve

of the velocity-time graph


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Example 1: Motion Diagrams

• Uniform velocity (shown by red arrows maintaining the

same size)

• Acceleration equals zero


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Example 2:

• Velocity and acceleration are in the same direction

• Acceleration is uniform (blue arrows maintain the same length)

• Velocity is increasing (red arrows are getting longer)


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Example 3:

• Acceleration and velocity are in opposite directions

• Acceleration is uniform (blue arrows maintain the same length)

• Velocity is decreasing (red arrows are getting shorter)


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Animation

O di i l M ti With C t t


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One-dimensional Motion With Constant

Acceleration

• If acceleration is uniform (i.e. ):

at vv o f

aa

t

vv

t t

vv

a

o f

f

o f

0

O di i l M ti With C t t


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One-dimensional Motion With Constant

Acceleration

• Used in situations with uniform acceleration

t vv

t v x f o

average

2

212

o x v t at Velocity changes

uniformly!!!

2 22 f ov v a x

at vv o f

General Motion with Constant


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General Motion with Constant

Acceleration

2

2

1

00

2

21

0

at t v x x

at t v x

at vvt 0

Summary: Motion under constant


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Summary: Motion under constant

acceleration


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Free Fall

• All objects moving under the influence of only

gravity are said to be in free fall

• All objects falling near the earth’s surface fall with

a constant acceleration

• This acceleration is called the acceleration due to

gravity, and indicated by g


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Acceleration due to Gravity

• Symbolized by g

• g = 9.8 m/s² (can use g = 10 m/s² for estimates)

• g is always directed downward – toward the center of the earth


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Free Fall -- an Object Dropped

• Initial velocity is zero

• Frame: let up be positive

• Use the kinematic equations

– Generally use y insteadof x since vertical

vo= 0

a = g

2

2

8.9

2

1

sma

at y

y

x

Free Fall -- an Object Thrown


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Free Fall -- an Object Thrown

Downward

• a = g

– With upward being positive,

acceleration will be negative, g =

-9.8 m/s²

• Initial velocity 0 – With upward being positive,

initial velocity will be negative

Free Fall -- object thrown


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Free Fall -- object thrown

upward

• Initial velocity is upward,so positive

• The instantaneous velocityat the maximum height is

zero• a = g everywhere in the

motion

– g is always downward,

negative

v = 0


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Thrown upward

• The motion may be symmetrical

– then tup = tdown

– then vf = -vo

• The motion may not be symmetrical – Break the motion into various parts

• generally up and down

Non-symmetrical


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Non-symmetrical

Free Fall

• Need to divide the

motion into segments

• Possibilities include

– Upward and downward

portions

– The symmetrical

portion back to the

release point and thenthe non-symmetrical

portion


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Combination Motions


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Projectile Motion


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Rules of Projectile Motion

• Introduce coordinate frame: y is up

• The x- and y-components of motion can be treatedindependently

• Velocities (incl. initial velocity) can be broken downinto its x- and y-components

• The x-direction is uniform motionax = 0

• The y-direction is free fall|ay|= g

S


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Some Details About the Rules

• x-direction – ax = 0

–

– x = vxot

• This is the only operative equation in the x-directionsince there is uniform velocity in that direction

constantvcosvv xooxo


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More Details About the Rules

• y-direction –

– take the positive direction as upward

– then: free fall problem• only then: ay = -g (in general, |ay|= g)

– uniformly accelerated motion, so the motion

equations all hold

ooy o sinvv


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Velocity of the Projectile

• The velocity of the projectile at any point of its

motion is the vector sum of its x and y components

at that point

x

y12y

2x

v

vtanandvvv

Maximum height reached


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Maximum height reached

Time taken for getting there

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Final velocity, v y 0

Height reached, hmax y y0

Using kinematics equation, v y2 v0 y

2 2 g y y0

hmax v0 y

2

2 g

Time taken to reach this height, using v y v0 y gt ,

t max v0 y

g

Depends only on the vertical component of the initial velocit

M i R


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Maximum Range

Total time of travel, t 2v0 y

g (twice the time to top)

Range is maximum distance traveled along horizontal axis

R v0 xt v0 x2v0 y

g v0 cos 0

2v0 sin 0 g

R v0

2 sin2 0

g

, using trig. id. sin2 2sin cos

Depends on both magnitude and direction of initial velocit

Maximum range is for sin2 1, i.e., 45o

E l f P j til M ti


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Examples of Projectile Motion:

• An object may be fired

horizontally

• The initial velocity is all in

the x-direction

– vo = vx and vy = 0

• All the general rules of

projectile motion apply

Non-Symmetrical Projectile


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Non Symmetrical Projectile

Motion

• Follow the general rulesfor projectile motion

• Break the y-direction intoparts

– up and down – symmetrical back to initial

height and then the rest ofthe height


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Example problem:

A rescue plane drops a package of

emergency rations to a stranded party of

explorers. The plane is traveling horizontally

at 40.0 m/s at a height of 100 m above the

ground.

Where does the package strike the ground

relative to the point at which it was released?

Given:

velocity: v=40.0 m/s

height: h=100 m

Find:

Distance d=?

2. Note: vox= v = + 40 m/s

voy= 0 m/s

1. Introduce coordinate frame:

Oy: y is directed up

Ox: x is directed right

2

2

1 2: ,

2

2 ( 100 ): 4.51

9.8

yOy y gt so t

g

mor t s

m s

m s sm x sot v xOx x 180)51.4)(40(,: 0

d

C T t 1


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ConcepTest 1

Consider the situation depicted here. A gun is accuratelyaimed at a dangerous criminal hanging from the gutter of abuilding. The target is well within the gun’s range, but theinstant the gun is fired and the bullet moves with a speed v o,the criminal lets go and drops to the ground. What happens?The bullet

1. hits the criminal regardlessof the value of v o.

2. hits the criminal only if v o is

large enough.

3. misses the criminal.

Equations for Constant x (meters)


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Equations for Constant

Acceleration

x = v0t + 1/2 at2 (parabolic)

v = at (linear)

v2 = v02 + 2a x (independent of time)

0

5

10

15

20

0 5 10 15 20

v (m/s)

t (seconds)

0

50

100

150

200

0 5 10 15 20

t (seconds)

0

0.5

1

1.5

2

0 5 10 15 20

a (m/s2)

t (seconds)

S f i t t t


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Summary of important concepts

• position

• displacement

• velocity

– average – instantaneous

• acceleration

– average

– instantaneous

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