75
Chapter 2: 1-D Kinematics
Chapter 2: 1-D Kinematics
Translational Motion Circular Motion
Projectile Motion Rotational Motion
Types of Motion
Natural Motion
•Objects have a proper place
•Objects seek their natural place
•External forces must be constantly
applied to moving objects in order
to keep them going.
•The heavier the object,
the faster it falls.
•Did not experiment to test theories.
Galileo Challenged The Dogma
Of Natural Motion with
Experiments
The natural motion of
a body is to remain in
whatever state of
motion it is in unless
acted upon by net
external forces.
Galileo Challenged The Dogma
Of Natural Motion
Galileo Challenged Aristotle Physics
In a vacuum, all objects fall with the same
acceleration due to gravity: 9.80 m/s2,
independent of their weight.
Galileo’s Motion Studies
0
2
f
xv
t
v vv
va
t
gave us…
Definitions:
Distance and Displacement
(delta) means "change in"
= 'final - initial'
The total distance traveled relative to an origin.
Distance is a scalar.
Displacement is a vector. The unit is the meter.
0fx x x
An ant zig-zags back and forth on a picnic table as shown.
The ant’s distance traveled and displacement are
A. 50 cm and 50 cm.
B. 30 cm and 50 cm.
C. 50 cm and 30 cm.
D. 50 cm and –50 cm.
E. 60 cm and –30 cm.
QuickCheck 2.1
Slide 2-29
An ant zig-zags back and forth on a picnic table as shown.
The ant’s distance traveled and displacement are
A. 50 cm and 50 cm.
B. 30 cm and 50 cm.
C. 50 cm and 30 cm.
D. 50 cm and –50 cm.
E. 60 cm and –30 cm.
QuickCheck 2.1
Slide 2-30
Average Speed &Velocity
Speed is how fast something moves.
The average speed is the total distance per time.
The average velocity is the the total displacement per time.
Velocity is a vector. The unit is m/s.
total displacement
total time
xv
t
Sense of Speed
1 / 3.6 / 2.24 /
10 / 36 / 22.4 /
20 / 72 / 44.8 /
30 / 108 / 67.2 /
m s km hr mi hr
m s km hr mi hr
m s km hr mi hr
m s km hr mi hr
1 / 2.25 /m s mi hr
1 / 0.62 /km hr mi hr
Acceleration How fast How fast is changing.
The rate at which the speed is changing.
Speeding up
Slowing down
Constant speed, changing direction.
change in velocity
change in time
va
t
Acceleration is in the direction of
the net Force but not necessarily
in the direction of velocity. Velocity is always in the direction of the motion!
Quicky Question
An automobile enters a freeway on-ramp at 15.0m/s and accelerates
uniformly up to 25.0 m/s in a time of 10.0s.
a) What is the automobile’s average velocity?
15 / 25 /20 /
2
m s m sm s
Which equation?
0
2
fv vv
An automobile enters a freeway on-ramp at 15.0m/s and accelerates
uniformly up to 25.0 m/s in a time of 10.0s.
b) What is the automobile’s average acceleration?
f iv vva
t t
225 / 15 /1 /
10
m s m sm s
s
Which equation?
Quicky Question
An automobile enters a freeway on-ramp at 15.0m/s and accelerates
uniformly up to 25.0 m/s in a time of 10.0s.
c) What is the distance traveled in this amount of time?
Which equation?
Quicky Question
200x m
(20 / )(10 )x
v x v t m s st
Motion Diagrams
Draw the Motion Diagram
An automobile enters a freeway on-ramp at 15.0m/s and accelerates
uniformly up to 25.0 m/s in a time of 10.0s.
Skiing through the woods
Here is a motion diagram of a car speeding up on a straight
road:
The sign of the acceleration ax is
A. Positive.
B. Negative.
C. Zero.
QuickCheck 2.13
Slide 2-69
Here is a motion diagram of a car speeding up on a straight
road:
The sign of the acceleration ax is
A. Positive.
B. Negative.
C. Zero.
QuickCheck 2.13
Slide 2-70
Speeding up means vx and ax have the same sign.
Which position-versus-time graph represents the motion shown in the
motion diagram?
Which position-versus-time graph represents the motion shown in the motion diagram?
Galileo’s Motion Studies
0 , ,
2
fv vx vv v a
t t
gave us…
Kinematic Equations
With a little al-jbr….
0 , ,
2
fv vx vv v a
t t
va
t
Start:
0fv v a t
Assume constant acceleration!
0fv v a t
0 , ,
2
fv vx vv v a
t t
va
t
Start:
0fv v a t
Assume constant acceleration!
0fv v a t
0 , ,
2
fv vx vv v a
t t
0
2
fv vx
t
Start:
0 0( )
2
f ix x v v a t
t
2
0
1
2f ix x v t a t
0 , ,
2
fv vx vv v a
t t
0
2
fv vx
t
0
2
f
t xv v
va
t
Start:
Combine &
Eliminate t:
0 =
fv vt
a
2 0
0
v vf
t xv v a
f
2 2
0 2fv v a x Algebra:
Galileo’s Motion Studies
0
0
2
0
2 2
0
, , 2
1
2
2
f
f
f
v vx vv v a
t t
v v at
x v t at
v v a x
gave us…
Kinematic Equations
Problem Solving Strategy
Acceleration: Changing Velocity From t = 0, how long does it take the car to come to a full stop?
How far does the car travel before it comes to a stop?
+x
Draw the Motion Graph
Acceleration: Changing Velocity
2
Knowns
5 /
28 /
0
?
i
f
a m s
v m s
v
t
f iv vt
a
2
0 28 /5.6
5 /
m ss
m s
f iv v at
Which equation to use?
Solve for t:
5.6t s
Acceleration: Changing Velocity
From t = 0, to t = 5.6s, how far does the car travel before it
comes to a stop?
+x
2
Knowns
5 /
28 /
0
5.6
i
f
a m s
v m s
v
t s
Which equation? 2
0
1
2x v t at
2 2128 5.6 ( 5 / )(5.6 ) 78.4
2
mx s m s s m
s
78.4x m
YOU TRY IT!
Brake Question You are driving a car going 80 km/hr (50mph) when a head-on
collision happens 25 meters ahead of you. If you can brake at 6 m/s2,
will you hit the crash or stop before it?
2 2
0 2fv v a x 2
0
2
vx
a
2
2
(22 / )40.3 25
2( 6 / )
m sx m m
m s
2
0: 80 / 22 / , 0, 6 / fKnowns v km hr m s v a m s
: ?Unknown x
CRASH!
Stopping
Distance
goes as the
SQUARE
of the
speed!
Stopping Distance
Traveling at 70 miles per hour, what is your breaking distance?
If v doubles,
d quadruples!!!
2
0
2
vx
a
Stopping
Distance
goes as the
SQUARE
of the
speed!
2 2
0 2fv v a x
Motion Graphs
What kind of motion is this?
What kind of motion is this?
xv
t
3
4001 /
400
mv m s
s
2 0 /v m s
1
4002 /
200
mv m s
s
What is the velocity during each
segment?
Slide 2-114
Galileo’s Motion Studies
0
0
2
0
2 2
0
, , 2
1
2
2
f
f
f
v vx vv v a
t t
v v at
x v t at
v v a x
gave us…
Definitions of averages
Kinematic Equations with constant acceleration
Constant vs Changing Acceleration
Depends on the FORCE
Constant Forces
• Constant pushes and pulls
• Inclined planes
• Gravity near the earth (Free Fall)
• Pulleys, Conical Pendulums
Variable Forces
• Springs and Pulleys
• Air Resistance
• Gravity Far from Earth
• Electricity and Magnetism
• MOST FORCES!!!!
Free Fall Unless told otherwise, ignore air resistance for
free fall problems!
Galileo Challenged Aristotle Physics
In a vacuum, all objects fall with the same
acceleration due to gravity: 9.80 m/s2,
independent of their weight.
Acceleration of Freely Falling Object
• The acceleration of an object in
free fall is directed downward,
regardless of the initial motion
• The magnitude of free fall
acceleration is g = 9.80 m/s2
g decreases with increasing altitude
– g varies with latitude
– 9.80 m/s2 is the average at the Earth’s
surface
– We will neglect air resistance
– g is a SCALAR!!! POSTIVE
Free Fall Equations For any object in the absence of air resistance.
29.80 /ya g m s
0
2
0
2 2
0
Customize:
1
2
2
f
f
v v gt
y v t gt
v v g y
0
2
0
2 2
0
Kinematic Eqs:
1
2
2
f
f
v v at
x v t at
v v a x
Note: v0 can be negative!
(taking up as +y)
Falling from Rest
2 215
2y at t
2
:
~ 10 /
Estimate
a g m s
10v at t
20 /
20
v m s
y m
10 /
5
v m s
y m
30 /
45
v m s
y m
40 /
80
v m s
y m
50 /
125
v m s
y m
+
0
2
0
1
2
fv v gt
y v t gt
0 0v
!v y How FAR is not
How FAST!
Take
down
as +y:
How Far: y(t) ~ t2
0fv v at
2
0
1
2y v t at
How Fast: v(t) ~ t1
+
How Fast How Fast is
Changing! 29.80 /g m s
FIRST: Define Reference Frame In this reference frame,what is the sign of a? 29.80 /a m s
What is v at t = 3s?
0fv v at
20 9.80 (3 )
ms
s
29.4m
s
2
0: 0, 9.8 / , 3Knowns v a m s t s
: ?fUnknown v
Negative because it is moving downward, in the negative direction!
FIRST: Define Reference Frame
2
0: 0, 9.8 / , 3 , 29.4 /fKnowns v a m s t s v m s
: ?Unknown y
The displacement is negative because it is moves downward, in the negative
direction but “how far” is a distance – a scalar – and is positive!
How far did the ball fall in those 3 seconds?
2
0
1
2y v t at
2
2
1
20 ( 9.8 )(3 )
ms
s
44.1m
The ball fell 44.1m.
Throwing up is Also Free Fall!
Symmetry of G Field.
2
:
~ 10 /
Estimate
a g m s
0
2
0
1
2
fv v gt
y v t gt
What Goes Up Must Come Down
Someone standing at the
edge of a cliff throws one
ball straight up and one
straight down at the same
speed. Ignoring air
resistance, which ball
strikes the ground with the
greatest speed?
Free Fall Question: You throw the rock down with an initial speed of
30 m/s. The rock hits the ground in 3 seconds. With what
speed will the rock hit the ground?
+y
230 9.8 (3 )
m ms
s s
59.4f
mv
s
0fv v at
How high is the cliff?
2
0: 30 / , 9.8 / , 3Knowns v m s a m s t s
: ?fUnknown v
Free Fall
2
0
1
2y v t at
2 21
2( 30 / )(3 ) ( 9.8 / )(3 )m s s m s s
134m
The cliff is 134 m high.
2
0: 30 / , 9.8 / , 3Knowns v m s a m s t s
: ?Unknown y +y
Question: You throw the rock down with an initial speed of
30 m/s. The rock hits the ground in 3 seconds. With what
speed will the rock hit the ground? How high is the cliff?
Free Fall: Throwing Up What is the speed at the top of the path?
ZERO!
What is the acceleration at the top?
a = -9.80 m/s2
What is the velocity at the same height
on the way down?
-30 m/s
+y
With what velocity will
the rock hit the ground?
-59.4 m/s
SAME as if you threw it
straight down at 30m/s!
How long does it take to hit the ground? First try to guess!
+y
0fv v at
0
2
59.4 / 30 /
9.8 /
fv v m s m st
a m s
9.12t s
2
0: 30 / , 9.8 / , 3 , 59.4 /fKnowns v m s a m s t s v m s
: ?Unknown t
How long to the top? How long back to launch point? Final v increases by 30m/s?
I guess about 9 seconds!
Free Fall: Throwing Up Problem
A ball is tossed straight up in the air. At its very
highest point, the ball’s instantaneous acceleration ay
is
A. Positive.
B. Negative.
C. Zero.
QuickCheck 2.18
Slide 2-96
A ball is tossed straight up in the air. At its very
highest point, the ball’s instantaneous acceleration ay
is
A. Positive.
B. Negative.
C. Zero.
QuickCheck 2.18
Slide 2-97
© 2013 Pearson Education, Inc.
Figure (a) shows the motion diagram of an object sliding down a straight, frictionless inclined plane.
Figure (b) shows the the free-fall acceleration the object would have if the incline suddenly vanished.
This vector can be broken into two pieces: and .
The surface somehow “blocks” , so the one-dimensional acceleration along the incline is
The correct sign depends on the direction the ramp is tilted.
Motion on an Inclined Plane
Slide 2-102
© 2013 Pearson Education, Inc.
The ball rolls up the ramp, then back down. Which is the correct acceleration graph?
© 2013 Pearson Education, Inc.
The ball rolls up the ramp, then back down. Which is the correct acceleration graph?
© 2013 Pearson Education, Inc.
Here is a motion diagram of a car moving along a straight road:
Which velocity-versus-time graph matches this motion diagram?
QuickCheck 2.5
Slide 2-44
© 2013 Pearson Education, Inc.
Here is a motion diagram of a car moving along a straight road:
Which velocity-versus-time graph matches this motion diagram?
QuickCheck 2.5
Slide 2-45
© 2013 Pearson Education, Inc.
Rank in order, from largest to smallest, the accelerations a
A– a
C at
points A – C.
A) aA > a
B > a
C
B) aA
> aC > a
B
C) aB
> aA > a
C
D) a
C > a
A > a
B
E) aC > aB > aA
© 2013 Pearson Education, Inc.
A) aA > a
B > a
C
B) aA
> aC > a
B
C) aB
> aA > a
C
D) a
C > a
A > a
B
E) aC > aB > aA
Rank in order, from largest to smallest, the accelerations a
A– a
C at
points A – C.
© 2013 Pearson Education, Inc.
Here is a position graph
of an object:
At t = 3.0 s, the object’s
velocity is
A. 40 m/s.
B. 20 m/s.
C. 10 m/s.
D. –10 m/s.
E. None of the above.
QuickCheck 2.7
Slide 2-50
© 2013 Pearson Education, Inc.
Here is a position graph
of an object:
At t = 3.0 s, the object’s
velocity is
A. 40 m/s.
B. 20 m/s.
C. 10 m/s.
D. –10 m/s.
E. None of the above.
QuickCheck 2.7
Slide 2-51
© 2013 Pearson Education, Inc.
Which velocity-versus-time graph or graphs goes with this acceleration-versus-time graph? The particle is initially moving to the right and eventually to the left.
© 2013 Pearson Education, Inc.
Which velocity-versus-time graph or graphs goes with this acceleration-versus-time graph? The particle is initially moving to the right and eventually to the left.
Which velocity-versus-time graph goes with the position-versus-time graph on the left?
Which velocity-versus-time graph goes with the position-versus-time graph on the left?
Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle’s position at ti = 0 s is xi = –10 m.
Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle’s position at ti = 0 s is xi = –10 m.
Speedy Sally
Speedy Sally, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 155 m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at 2.00 m/s2 because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sally's car and the van. Sketch the x-t graphs for both the vehicles. What does it mean?
Rock Drop
A rock is dropped from rest into a well. The sound of the splash is heard 3.20 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s.
