Post on 07-Jul-2018
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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
49
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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