Chapter 2 Kinematics in One Dimension. Kinematics –Kinema is Greek for “motion” –Kinematics → Describing motion Kinematics describes motion in terms of.
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Chapter 2
Kinematics
in One Dimension
• Kinematics– Kinema is Greek for “motion”– Kinematics → Describing motion
• Kinematics describes motion in terms of – position– displacement– velocity (average and instantaneous)– acceleration
• Dynamics– Describing how interactions can change motion
• Mechanics– Kinematics and dynamics.
For one-dimensional motion, plus (+) and minus (-) are used to specify the vector direction. Plus (+) is at 0° along the positive x axis and minus (-) is at 180° along the negative x axis.
Position x is the location of the object
initial position final position
positive xdirection
x axis at 0°
position initial ox
position final x
ntdisplaceme oxxx
Change Δ always means (final value) – (initial value)
Displacement Δx is the change in position
x= 0
2 mo x
7.0 mx
5 m x
Example: Displacement in the positive x direction
initial Displacement is the change
final
7 m 2 m 5 mo x x x
ntdisplaceme oxxx
Displacement is +5 m
Example: Displacement in the negative x direction
2 mx
7 mo x
5 m x
initial
Displacement is the changefinal
2 m 7 m 5 mo x x x
Displacement is - 5 m
ntdisplaceme oxxx
Average velocity is the displacement divided by the elapsedtime.
DisplacementAverage velocity
Elapsed time
ttt o
o
xxx
v
SI units for velocity: meters per second
Velocity is the displacement in one second.
v "bar" means average
m
s
Example 2 The World’s Fastest Jet-Engine CarAndy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. The driver makes two runs through the course, one in each direction, to adjust for wind effects. Determine the average velocity for each run.
1609 m m339.5
4.740 s st
x
v
1609 m m342.7
4.695 s st
x
v
Vector directions are shown with +/- signs.
timeElapsed
Distance speed Average
SI units for speed: meters per second
In this course, speed will mean the magnitude of the velocity.
Ignore the textbook definition of speed
textbook definition:
m
s
Instantaneous velocity is the average velocity for a very short time interval.
0 0lim = limt t t
x
v v
average velocity
instantaneous velocity
Average velocity is the average during some time interval.
Instantaneous velocity is the velocity at a specific moment.
ttt o
o
vvv
a
Average acceleration is the velocity change in one second.
Velocity changeAverage acceleration
Elapsed time
"bar" means average
Example 3 Increasing velocity: Determine the average acceleration.
0 km/ho v
260km hv
s 0ot s 29t
260 09
29 s 0 s so
o
km km kmh h h
t t
v va
260km hv
s 29t0 km/ho v
s 0ot
v
a
v and a are in the same direction so v increases
Acceleration is the velocity change each second.
kmVelocity change is +9 each second.
h
v
a
v and a are in the same direction so v increases
Acceleration is the velocity change each second
mVelocity change is -5 each second.
s
v
a
v and a are in opposite directions so v decreases
8 m mVelocity = Slope 4
2 s s
x
t
Graphical representation of kinematic quantities
Position graph with a constant slope
Position graph with three different slopes
2m
vs
0m
vs
1m
vs
Velocity = Slope
Position graph with a variable slope
Velocity = Slope
x
t
265.2
5
x m mv
t s s
2
m12 msAcceleration = Slope 62 s s
v
t
Velocity graph with a constant slope
Your success in physics critically depends on your ability to correctly distinguish and correctly use technical terms as the terms are defined in physics.
velocity and acceleration are different physical quantities with different units
velocity is position change each second
acceleration is velocity change each second
velocity is different from velocity change
1. position (at time t), x
2. initial velocity (at time t=0), vo
3. acceleration (constant), a
4. final velocity (at time t), v
5. final time, t
The kinematic equations relate these 5 variables for situations where the acceleration is constant and allow us to solve for two of these quantities that are unknown.
Five kinematic variables
Five kinematic equations for constant acceleration cases
tvvx o 21
221 attvx o
atvv o
axvv o 222
x vt
Position equations Velocity equations
Use these equations to solve for any 2 missing variables.
You need to know 3 of the 5 variables to find the missing 2 !
Obtained from the other equations by eliminating t.
Derivation of the kinematic equations is shown at the end of the slides.
Success Strategy [ Very important steps !! ]
1. Make a drawing.2. Draw direction arrows. (displacement, velocity, acceleration)3. Label the positive (+) and negative (-) directions.4. Label all know quantities. 5. Use a table to organize known values with correct units. 6. Make sure you have three of the five kinematic quantities.7. Select the appropriate equations and solve.
There may be two possible answers because quadratic equations have 2 roots.
For multi-segmented motion, remember that the final velocity of one segment is the initial velocity for the next segment.
Example:
Find the final position xv
a same directions
so v increases
x a v vo t
m s
? ? 8s
2
m
sm
s
m
s
22m
s 6
m
s
212
212 2
2
6 8 s 2 8 s
112 m
6 2 8 s
m22
s
o
o
x v t at
m mx
s s
x
m mv v at
s s
v
x a v vo t
m s
? ? 8s
2
m
sm
s
m
s
22m
s 6
m
s
Example 6 Airplane take-off
Find the final position x
??x
v
a
same directions so v increases
x a v vo t
m s
? 0 ?
2
m
sm
s
m
s
231m
s 62
m
s
0
2
221 12 2 2
m62 0 31
s
2
m0 2 31 2 62
so
v v at
m mt
s s
t s
mx v t at s s m
s
Example 6 Airplane take-off
x a v vo t
m s
? 0 ?
2
m
sm
s
m
s
231m
s 62
m
s
In the absence of air resistance, all bodies at the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is small compared to the radius of the Earth, then the acceleration remains constant throughout the motion.
This idealized motion is called free-fall and the accelerationof a freely falling body is called the acceleration due to gravity. The direction of this acceleration is downward.
29.8m
gs
The magnitude of the acceleration is called "g".
Free fall -- motion only influenced by gravity
2
2
9.8
9.8
mg
sm
as
downward
y axis positive upwardnegative downward
a
y
air resistance influencesfalling
no air (vacuum)(not vacuum cleaner)
air
v
v
A stone is dropped from the top of a tall building. What is the displacement y of the stone after 3 s of free fall?
y a v vo tm s
? -9.8 m/s2 ? 0 m/s 3 s
2
m
sm
s
m
s
212
212 2
0 3 s 9.8 3 s
44.1 m
oy v t at
m my
s s
y
a
y a v vo t
? ? ? +5 m/s ?
A coin is tossed upward with an initial speed of 5 m/s. How high does the coin go above its release point? Ignore air resistance.
Only one value is given in the wording of this problem. You are expected to add two values by thinking.
Vertical velocity always momentarily zero at the top .
v
a
y a v vo t
? -9.8 m/s2 0 m/s +5.00 m/s ?
2
2
2
m m m0 5 9.8
s s s
m m5 9.8
s sm
5s 0.51m
9.8s
ov v at
t
t
t s
212
212 2
m m5 0.51 9.8 0.51
s s
2.55 1.27 1.28
oy v t at
y s s
y m m m
Acceleration versus Velocity
There are three parts to the motion of the coin.
On the way up, the vector velocity is upward and the vector acceleration is downward so the velocity change is downward (v gets less upward).
At the top, the vector velocity is momentarily zero and the vector acceleration is downward so the velocity change is downward (v gets more downward).
On the way down, the vector velocity is downward and the acceleration vector is downward so the velocity change is downward (v gets more downward).
The velocity changes, but does the acceleration change?
v a
v a
0 v a
The End
Derivation of the kinematic equations
o
o
tt
xxv
0ox 0ot
x vt
We assume the object is at the origin at time to = 0.
t
xv
12 ov v v Also, for constant acceleration cases
position equation
and
Deriving the kinematic equations ......
Velocity definition
multiply both sides by t
o
o
tt
vva
t
vva o
ovvat
atvv o
0ot
Velocity equation
Acceleration definition
multiply both sides by t
1 12 2
12
212
o
o o o
o o
o
v v at
v v v v v at
x vt
x v v at t
x v t at
another position equation
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