Physics 121,Lecture 4,Pg 1
Physics 121,Sections 9,10,11,and 12
Lecture 4
Announcements
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Physics 121,Lecture 4,Pg 2
Physics 121,Sections 9,10,11,and 12
Lecture 3
Today?s Topics:
? Homework 1,Due Friday Sept,6 @ 6:00PM
? Ch.1,# 4,10,14,19,31,and 33.
? Ch.2,# 1,23,37,and 63.
? Chapter 3,Forces and motion along a line
? Position
? Velocity,average & instantaneous
? Acceleration,average & instantaneous
? Force and acceleration
? Motion with constant acceleration
? Falling objects
? Apparent weight
Physics 121,Lecture 4,Pg 3
Chapter 3,Position / Displacement
? Displacement is just change in position.
? ?X = xf - xi
? Displacement is a vector quantity
? A vector quantity has both magnitude and direction
? A scalar quantity has only magnitude and no direction
10 meters 15 meters
O
xi Joexf
?x = 5 meters
{
Physics 121,Lecture 4,Pg 4
Average Velocity
? The average velocity v during the time interval ?t is
defined as the displacement ?x divided by ?t.
if
if
tt
xx
t
xv
?
?
?
?
??
Physics 121,Lecture 4,Pg 5
Instantaneous Velocity
? The instantaneous velocity v is defined as the
limit of the average velocity as the time interval ?t
becomes infinitesimally short
t
xv
t ?
??
?? 0
lim
Physics 121,Lecture 4,Pg 6
Average Acceleration
? The average acceleration a during the time
interval ?t,is defined as the change in velocity ?v
divided by the ?t
? It is a vector …
if
if
tt
vv
t
va
?
?
?
?
??
Physics 121,Lecture 4,Pg 7
Instantaneous Acceleration
? The instantaneous acceleration a is defined as
the limit of the average acceleration as the time
interval ?t goes to zero
t
va
t ?
??
?? 0
lim
Physics 121,Lecture 4,Pg 9
Kinematic Variables
? Measured with respect to a reference frame.
(x-y axis)
? Measured using coordinates (having units).
? Many kinematic variables are vectors,which
means they have a direction as well as a
magnitude.
? Vectors denoted by boldface v or arrow ?v
Physics 121,Lecture 4,Pg 10
Newton’s 2nd law and acceleration
? It relates the net force to the acceleration
? A force is what changes the velocity of a particle
?No net force,no change in velocity
? The constant of proportionality is the mass
?For a given Fnet a more massive object will undergo a
smaller acceleration (and vice versa)
Physics 121,Lecture 4,Pg 11
Motion in 1 dimension
? In 1-D,we usually write position as x,
? Since it?s in 1-D,all we need to indicate direction is +
or ?.
? Displacement in a time ?t = tf - ti is ?x = xf - xi
t
x
ti tf
??x
??t
xi
xf some particle?s trajectoryin 1-D
Physics 121,Lecture 4,Pg 12
1-D kinematics
t
x
tt
xxv
if
if
av ?
??
?
??
? Velocity v is the,rate of change of position”
? Average velocity vav in the time ??t = tf - ti is:
t
x
t1 t2
??x
x
1
x
2
trajectory
??t
Vav = slope of line connecting x1 and x2.
Physics 121,Lecture 4,Pg 13
? Consider limit tf - ti ? 0
? Instantaneous velocity v is defined as:
1-D kinematics...
t
xtv
t ?
??
?? 0
lim)(
t
x
t1 t2
??x
x
1
x
2
??t
so v(t2 ) = slope of line tangent to path at t2.
Physics 121,Lecture 4,Pg 14
1-D kinematics...
? Acceleration a is the,rate of change of velocity”
? Average acceleration aav in the time ??t = t2 - t1
is,
t
v
tt
vva
if
if
av ?
??
?
??
? If a = aav is constant,and set t1=0,v1= v0,and
then v2= v
?
Physics 121,Lecture 4,Pg 15
Average Velocity
? Velocity is,v = v0 + at (for constant acceleration)
v
t
t
v
vav
v0
? ?vv21v 0av ??
Physics 121,Lecture 4,Pg 16
1-D kinematics
? We saw that
v = ?x / ? t
After a little algebra we have:
?x = v ?? t
?Graphically,we can add up lots of small rectangles:
x(t)
t
+ +...+
= displacement
Physics 121,Lecture 4,Pg 17
1-D kinematics
? For constant acceleration a we find,? ?vv21v 0av ??
? Recall, vav = ?x / ? t
? Again,set t1=0 and t2=t so that ? t = t
? And so
Physics 121,Lecture 4,Pg 18
Recap
? So for constant acceleration we find:
t
v v at? ?0 v t
ta c o n s t? a
x x v t at? ? ?0 0 212x
Physics 121,Lecture 4,Pg 19
Derivation:
? Plugging in for t:
atvv 0 ?? 200 at21tvxx ???
? Solving for t:
a
vvt 0??
2
0000 a vva21a vvvxx ?
?
??
?
? ???
?
??
?
? ???
)xx(a2vv 0202 ???
Physics 121,Lecture 4,Pg 20
Recap:
? For constant acceleration:
atvv 0 ??
200 at21tvxx ???
a c o n s t?
? From which we know:
v)(v21v
)x2a ( xvv
0av
0
2
0
2
??
???
Active Figure 2
Physics 121,Lecture 4,Pg 21
Lecture 4,Act 1
Motion in One Dimension
? When throwing a ball straight up,which of the following is
true about its velocity v and its acceleration a at the
highest point in its path?
(a) Both v = 0 and a = 0.
(b) v ? 0,but a = 0.
(c) v = 0,but a ? 0.
y
Physics 121,Lecture 4,Pg 22
Free Fall
? When any object is let go it falls toward the ground !!
The force that causes the objects to fall is called
gravity.
? The acceleration caused by gravity is typically
written as g
? Any object,be it a baseball or an elephant,
experiences the same acceleration (g) when it is
dropped,thrown,spit,or hurled,i.e,g is a constant,
Physics 121,Lecture 4,Pg 23
Gravity facts:
? g does not depend on the nature of the material!
?Galileo (1564-1642) figured this out without fancy clocks
& rulers!
? demo - feather & penny in vacuum
? Nominally,g = 9.81 m/s2
?At the equator g = 9.78 m/s2
?At the North pole g = 9.83 m/s2
? More on gravity in a few lectures!
Physics 121,Lecture 4,Pg 24
Lecture 4,Act 2
? A car is traveling with an initial velocity vo,At t = 0,
the driver puts on the brakes,which slows the car at a
rate of ab
x = 0,t = 0ab
vo
Physics 121,Lecture 4,Pg 25
Lecture 4,Act 2
? A car traveling with an initial velocity vo,At t = 0,the
driver puts on the brakes,which slows the car at a
rate of ab,At what time tf does the car stop,and how
much farther xf does it travel
x = xf,t = tf
v = 0
x = 0,t = 0ab
v0
Physics 121,Lecture 4,Pg 26
Problem:
? On a bright sunny day you are walking around
the campus watching one of the many
construction sites,To lift a bunch of bricks from
a central area,they have brought in a helicopter,
As the pilot is leaving,she accidentally releases
the bricks when they are 1000 m above the
ground,The worker below is getting ready to
walk away in 10 seconds,Does he live?
Physics 121,Lecture 4,Pg 27
Problem Solution Method:
Five Steps:
1) Focus the Problem
- draw a picture – what are we asking for?
2) Describe the physics
- what physics ideas are applicable
- what are the relevant variables known and unknown
3) Plan the solution
- what are the relevant physics equations
4) Execute the plan
- solve in terms of variables
- solve in terms of numbers
5) Evaluate the answer
- are the dimensions and units correct?
- do the numbers make sense?
Physics 121,Lecture 4,Pg 28
Problem:
1,We need to find the time it
takes for the brick to hit the
ground.
1000 m
Physics 121,Lecture 4,Pg 29
Problem:
2,Free Fall
Constant acceleration at value g
? Choose coordinate system
? Variables:
h – height of helicopter = 1000 m
g – acceleration due to gravity = 9.81 m/s2
t – time to drop – goal
m – mass of brick – unknown
vo – initial velocity = 0 why?
1000 m
y = 0
y
Physics 121,Lecture 4,Pg 30
Problem:
3,Next write position for
constant acceleration:
20y0 gt
2
1tvyy ??
1000 m
y = 0
y
Realize that v0y = 0
Solve for when y=0?
Why
Physics 121,Lecture 4,Pg 31
Problem:
y0 = 1000 m
y
y = 0
g
yt 02?
Solve for t in symbols:
20 gt
2
1yy ??4.Substitute y=0 and vo=0
s314sm819 m10 002gy2t 20,,????
Solve for t in numbers:
Physics 121,Lecture 4,Pg 32
Problem:
y0 = 1000 m
y
y = 0
5,Does it make sense?
? Units worked out to be ?s?
That?s correct for time
? It takes 14.3 s.
That means that the man
escapes !!!
Physics 121,Lecture 4,Pg 33
Problem #2
? As you are driving to school one day,you pass a
construction site for a new building and stop to watch for a
few minutes,A crane is lifting a batch of bricks on a pallet
to an upper floor of the building,Suddenly a brick falls off
the rising pallet,You clock the time it takes for the brick to
hit the ground at 2.5 seconds,The crane,fortunately,has
height markings and you see the brick fell off the pallet at a
height of 22 meters above the ground,A falling brick can
be dangerous,and you wonder how fast the brick was
going when it hit the ground,Since you are taking physics,
you quickly calculate the answer.
?a,Draw a picture illustrating the fall of the brick,the length it
falls,and the direction of its acceleration.
?b,What is the problem asking you to find?
?c,What kinematics equations will be useful?
?d,Solve the problem in terms of symbols.
?e,Does you answer have the correct dimensions?
?f,Solve the problem with numbers.
Physics 121,Lecture 4,Pg 34
Solution / Problem #2
a,Draw a diagram
b,What do you need to calculate
c,Which kinematics equations will be useful?
h = 22 m
brick
floor
t = 2.5 s g
vo
acceleration in free fall (a = const = g ~ 10 m/s2)
Speed of the brick when it hits the ground (v)
atvv 0 ??
200 at21tvxx ??? 0= h +v0 t - g t2 / 2
v = v0 - g t
Physics 121,Lecture 4,Pg 35
Solution / Problem #2 (cont.)
d,Solve the problem in terms of symbols.
e,Does you answer have the correct dimensions (what are they)?
f,Solve the problem with numbers.
= - 21.3 m /s
h = v0 t - g t2 / 2
v = v0 - g t
1
2
1
tv0 = ( g t
2 - h )
1
2
1
tv = ( g t
2 - h ) - g t
{ }v = 121 [ T ] [ L / T 2 ] ?[?T?]?2 - [ L ] - [ L / T 2 ] ?[?T?]
{ }v = 121 2.5 s 10 m /s2 ? (2.5 s )2 - 22 m - 10 m /s2 ? 2.5 s
Physics 121,Lecture 4,Pg 36
Lecture 4,ACT 3
1D Freefall
? Alice and Bill are standing at the top of a cliff of height
H,Both throw a ball with initial speed v0,Alice straight
down and Bill straight up,The speed of the balls when
they hit the ground are vA and vB respectively.
v0
v0
BillAlice
H
vA vB
Which of the following is true:
(a) vA < vB (b) vA = vB (c) vA > vB
Physics 121,Lecture 4,Pg 37
Recap of today’s lecture
? Chapter 3,Forces and motion along a line
?Position
?Velocity,average & instantaneous
?Acceleration,average & instantaneous
?Force and acceleration
?Motion with constant acceleration
?Falling objects
? Homework 1 on WebAssign