Physics 121,Lecture 6,Pg 1
Physics 121,Sections 9,10,11,and 12
Lecture 6
Today’s Topics:
? Homework 2,Due Friday Sept,16 @ 6:00PM
? Ch.3,# 2,11,18,20,25,32,36,46,50,and 56.
? Chapter 4,Motion in 2-D
? Review of vectors
? Projectile motion
? Relative velocity
?More examples of FBD’s
Physics 121,Lecture 6,Pg 2
Katzenstein Distinguished Lecture
? Prof,Franck Wilczek
? Nobel Prize in Physics 2004
? From MIT
? Title,The Universe is a Strange Place
? Where,Room P-36
? When,Friday at 4:00PM
? Refreshments at 3:00 PM in front of P-36
? Come for a great talk
Physics 121,Lecture 6,Pg 3
Unit Vectors:
? A Unit Vector is a vector having length 1
and no units.
? It is used to specify a direction.
? Unit vector u points in the direction of U.
?Often denoted with a,hat”,u = ?
U
?
x
y
z
i
j
k
? Useful examples are the cartesian
unit vectors [ i,j,k ]
?point in the direction of the
x,y and z axes.
R = rxi + ryj + rzk
Physics 121,Lecture 6,Pg 4
Vector addition using components:
? Consider C = A + B.
(a) C = (Ax i + Ay j ) + (Bx i + By j ) = (Ax + Bx )i + (Ay + By )j
(b) C = (Cx i + Cy j )
? Comparing components of (a) and (b):
? Cx = Ax + Bx
? Cy = Ay + By C
BxA
ByB
Ax
Ay
Physics 121,Lecture 6,Pg 5
Lecture 6,ACT 1
Vector Addition
? Vector A = {0,2}
? Vector B = {3,0}
? Vector C = {1,-4}
What is the resultant vector,D,from
adding A+B+C?
(a) {3,-4} (b) {4,-2} (c) {5,-2}
Physics 121,Lecture 6,Pg 6
Review (1-D):
? For constant acceleration we found:
x
a
v t
t
t
v v at? ?0
200
2
1 attvxx ???
consta ?
? A few other useful formulas,
)x2 a ( xvv
v)(v
2
1v
0
2
0
2
0av
???
??
vav
Physics 121,Lecture 6,Pg 7
2-D Kinematics
? For 2-D,we simply apply the 1-D equations to each
of the component equations.
t
va x
x ?
??
t
xv
x ?
??
t
yv
y ?
??
t
va y
y ?
??
if xxx ??? if yyy ???
? Which can be combined into the vector equations:
??
?r ? r f ? r i
??
v ? ?r ?t
??
a ??v ?t
Physics 121,Lecture 6,Pg 8
2-D Kinematics
? So for constant acceleration we get:
?a = const
?v = v0 + a t
?r = r0 + v0 t + 1/2 a t2
(where a,v,v0,r,r0,are all vectors)
Physics 121,Lecture 6,Pg 9
3-D Kinematics
? Most 3-D problems can be reduced to 2-D problems when
acceleration is constant;
?Choose y axis to be along direction of acceleration.
?Choose x axis to be along the,other” direction of
motion.
? Example,Throwing a baseball (neglecting air resistance).
?Acceleration is constant (gravity).
?Choose y axis up,ay = -g.
?Choose x axis along the ground in the direction of the
throw.
Physics 121,Lecture 6,Pg 10
“x” and,y” components of motion are
independent.
? A man on a train tosses a ball straight up in the air.
?View this from two reference frames:
Reference frame
on the ground.
Reference frame
on the moving train.
y motion,a = -g y
x motion,x = v0t
Physics 121,Lecture 6,Pg 11
Projectile Motion.
? If I set something moving near the earth,it reduces to a 2-D
problem we call projectile motion,
? Use a coordinate system with x along the ground,y vertical
with respect to the ground,(Notice no change in third
direction.)
? Equations of motion reduce to:
?X,?x = voxt ax = 0
?Y,y = yo + voyt – g t2 /2 y positive upwards
Physics 121,Lecture 6,Pg 12
Lecture 6,ACT 2
2-D Motion
? Alice and Bill are playing air hockey on a table with no
bumpers at the ends,Alice scores a goal and the puck goes
flying off the end of the table,Which diagram best describes
the path of the puck?
Alice Bill
A) B) C)
Physics 121,Lecture 6,Pg 13
Problem:
? Sammy Sosa clobbers a fastball toward center-field,You
are checking out your new fancy radar gun which can detect
ball velocity,i.e,speed and direction,You measure that the
ball comes off the bat with initial velocity is 36.5 m/s at an
angle of 30o above horizontal,Since Sammy was hitting a
high fastball,you estimate that he contacted the ball about
one meter off of the ground,You know the dimensions of
Wrigley field and the center-field wall is 371 feet (113m) from
the plate and is 10 feet (3m) high,You decide to
demonstrate your superfast math and physics skills by
predicting whether Sammy get a home run before the play is
decided,
Physics 121,Lecture 6,Pg 14
Problem:
1) We need to find how high the ball is at a distance of 113m
away from where it starts,
?
v
h
D
yo
Physics 121,Lecture 6,Pg 15
Problem:
2) This is a problem in projectile motion.
Choose y axis up.
Choose x axis along the ground in the direction of the hit.
Choose the origin (0,0) to be at the plate.
Say that the ball is hit at t = 0,x = xo = 0,y = yo = 1m
?
v
h
D
y
x
Physics 121,Lecture 6,Pg 16
Problem...
? Variables
?vo = 36.5 m/s
?yo = 1 m
?h = 3 m
??o = 30o
?D = 113 m
?a = (0,ay) ? ay = -g
?t = unknown,
?Yf – height of ball when x=113m,unknown,
our target
Physics 121,Lecture 6,Pg 17
Problem...
3) For projectile motion,
? Equations of motion are:
vx = v0x vy = v0y - g t
x = vx t y = y0 + v0y t - 1/ 2 g t2
And,use geometry to find vox and voy
y
x
g
?
v
v0x
v0yy
0
Find v0x = |v| cos ?.
and v0y = |v| sin ?.
Physics 121,Lecture 6,Pg 18
Problem...4) Solve the problem,
? The time to reach the wall is,t = D / vx (easy!)
? Height at any time,y(t) = y0 + v0y t - g t2/ 2
? Combining the two gives,y(t) = y0 + v0y D/vox - g D2/ (2vox2)
? And substitute for vox and voy,y(t) = y0 + D tan? - g D2/ 2(vocos?)2
? All are known quantities,Solved.
? Numbers:
? y(t) = (1.0 m) + (113 m)(tan 30) -
(0.5)(9.8 m/s2)(113 m)2/(36.5 m/s cos 30)2
= (1.0 + 65.2 - 62.6) m = 3.6 m
Physics 121,Lecture 6,Pg 19
Problem...
5)Think about the answer,
? The units work out correctly for a height (m)
? It seems reasonable for the ball to be a little over 3m
high when it gets to the fence,
? But,we haven’t yet answered the question
? Since the wall is 3m high,and the ball is 3.26m high
when it gets there,Sammy gets a homer.
Physics 121,Lecture 6,Pg 20
? Two footballs are thrown from the same point on
a flat field,Both are thrown at an angle of 30o
above the horizontal,Ball 2 has twice the initial
speed of ball 1.
If ball 1 is caught a distance D1 from the thrower,
how far away from the thrower D2 will the
receiver of ball 2 be when he catches it?
(a) D2 = 2D1 (b) D2 = 4D1 (c) D2 = 8D1
Lecture 6,ACT 3
Motion in 2-D
Physics 121,Lecture 6,Pg 21
Shooting the Monkey
(tranquilizer gun)
? Where does the zookeeper
aim if he wants to hit the monkey?
( He knows the monkey will
let go as soon as he shoots ! )
Physics 121,Lecture 6,Pg 22
Shooting the Monkey...
? If there were no gravity,simply aim
at the monkey
r = r0
r =v0t
Physics 121,Lecture 6,Pg 23
Shooting the Monkey...
r = v0 t - 1/2 g t2
? With gravity,still aim at the monkey! r = r0 -
1/2 g t2
Dart hits the
monkey!
Physics 121,Lecture 6,Pg 24
Recap:
Shooting the monkey...
x = x0
y = -1/2 g t2
? This may be easier to think about,
It’s exactly the same idea!!
x = v0 t
y = -1/2 g t2
Physics 121,Lecture 6,Pg 25
Typical questions,
(projectile motion; for given v0 and ?)
? What is the maximum height the ball reaches (h)?
? How long does it take to reach maximum height?
h = (v0 sin ?) t - 1/2 g t2
v = (v0 sin ?) - g t = 0 at P
t = (v0 sin ?) / g
t = (v0 sin ?) / g !
y,
?
h
L
y
x
v0
P
? Would the answers above be any different if the
projectile was moving only along y-axis (1-D motion)
with the initial velocity,v0 sin (?)?
( A ) YES ( B ) NO ( C ) CAN’T TELL
h
y
xv0 sin(?)
P
Physics 121,Lecture 6,Pg 26
Typical questions,
(projectile motion; for given v0 and ?)
? What is the range of the ball (L)?
? How long does it take for ball to reach final point (P)?
?
h
L
y
x
v0
P
y = (v0 sin ?) t - 1/2 g t2 = 0 ! when at P
[ (v0 sin ?) - 1/2 g t] t = 0
t = 0 ; t = 2 (v0 sin ?) / g
L = vx0 t = (v0 cos ?) tx,
y,
Physics 121,Lecture 6,Pg 27
Problem 2
? Suppose a projectile is aimed at a target at rest placed at
the same height,At the time that the projectile leaves the
cannon the target is released from rest and starts falling
toward ground,Would the projectile miss or hit the target?
t = t1
y
xv0
t = 0
t = 0
TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Physics 121,Lecture 6,Pg 28
Problem 3
? Suppose a projectile is aimed at a target at rest somewhere
above the ground as shown in Fig,below,At the same time
that the projectile leaves the cannon the target falls toward
ground,Would the projectile miss or hit the target?
t = t1
?
y
x
v0
t = 0
t = 0
TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Physics 121,Lecture 6,Pg 29
Inertial Reference Frames:
? A Reference Frame is the place you measure from.
?It’s where you nail down your (x,y,z) axes!
? An Inertial Reference Frame (IRF) is one that is not
accelerating.
?We will consider only IRF’s in this course.
? Valid IRF’s can have fixed velocities with respect to each other,
?More about this later when we discuss forces.
?For now,just remember that we can make measurements
from different vantage points.
Physics 121,Lecture 6,Pg 30
Lecture 6,ACT 4
Relative Motion
? Consider an airplane flying on a windy day.
? A pilot wants to fly from New Haven to Bradley airport,
Having asked a friendly physics student,she knows that
Bradley is 120 miles due north of New Haven and there is
a wind blowing due east at 30 mph,She takes off from
New Haven Airport at noon,Her plane has a compass and
an air-speed indicator to help her navigate,She uses her
compass at the start to aim her plane north,and her air
speed indicator tells her she is traveling at 120 mph with
respect to the air,
After one hour,
A) She is at Bradley
B) She is due east of Bradley
C) She is southeast of Bradley
Physics 121,Lecture 6,Pg 31
Lecture 6,ACT 5
Relative Motion
? You are swimming across a 50m wide river in which the current
moves at 1 m/s with respect to the shore,Your swimming speed is
2 m/s with respect to the water,
You swim across in such a way that your path is a straight
perpendicular line across the river.
?How many seconds does it take you to get across?
2m/s
1m/s50m
s25250 ?a)
s50150 ?b)
s29350 ?c)
s35250 ?d)
Physics 121,Lecture 6,Pg 32
Recap of today’s lecture
? Homework 2,Due Friday Sept,16 @ 6:00PM
?Ch.3,# 2,11,18,20,25,32,36,46,50,and 56.
? Chapter 4,Motion in 2-D
?Review of vectors
?Projectile motion
?Relative velocity
Physics 121,Sections 9,10,11,and 12
Lecture 6
Today’s Topics:
? Homework 2,Due Friday Sept,16 @ 6:00PM
? Ch.3,# 2,11,18,20,25,32,36,46,50,and 56.
? Chapter 4,Motion in 2-D
? Review of vectors
? Projectile motion
? Relative velocity
?More examples of FBD’s
Physics 121,Lecture 6,Pg 2
Katzenstein Distinguished Lecture
? Prof,Franck Wilczek
? Nobel Prize in Physics 2004
? From MIT
? Title,The Universe is a Strange Place
? Where,Room P-36
? When,Friday at 4:00PM
? Refreshments at 3:00 PM in front of P-36
? Come for a great talk
Physics 121,Lecture 6,Pg 3
Unit Vectors:
? A Unit Vector is a vector having length 1
and no units.
? It is used to specify a direction.
? Unit vector u points in the direction of U.
?Often denoted with a,hat”,u = ?
U
?
x
y
z
i
j
k
? Useful examples are the cartesian
unit vectors [ i,j,k ]
?point in the direction of the
x,y and z axes.
R = rxi + ryj + rzk
Physics 121,Lecture 6,Pg 4
Vector addition using components:
? Consider C = A + B.
(a) C = (Ax i + Ay j ) + (Bx i + By j ) = (Ax + Bx )i + (Ay + By )j
(b) C = (Cx i + Cy j )
? Comparing components of (a) and (b):
? Cx = Ax + Bx
? Cy = Ay + By C
BxA
ByB
Ax
Ay
Physics 121,Lecture 6,Pg 5
Lecture 6,ACT 1
Vector Addition
? Vector A = {0,2}
? Vector B = {3,0}
? Vector C = {1,-4}
What is the resultant vector,D,from
adding A+B+C?
(a) {3,-4} (b) {4,-2} (c) {5,-2}
Physics 121,Lecture 6,Pg 6
Review (1-D):
? For constant acceleration we found:
x
a
v t
t
t
v v at? ?0
200
2
1 attvxx ???
consta ?
? A few other useful formulas,
)x2 a ( xvv
v)(v
2
1v
0
2
0
2
0av
???
??
vav
Physics 121,Lecture 6,Pg 7
2-D Kinematics
? For 2-D,we simply apply the 1-D equations to each
of the component equations.
t
va x
x ?
??
t
xv
x ?
??
t
yv
y ?
??
t
va y
y ?
??
if xxx ??? if yyy ???
? Which can be combined into the vector equations:
??
?r ? r f ? r i
??
v ? ?r ?t
??
a ??v ?t
Physics 121,Lecture 6,Pg 8
2-D Kinematics
? So for constant acceleration we get:
?a = const
?v = v0 + a t
?r = r0 + v0 t + 1/2 a t2
(where a,v,v0,r,r0,are all vectors)
Physics 121,Lecture 6,Pg 9
3-D Kinematics
? Most 3-D problems can be reduced to 2-D problems when
acceleration is constant;
?Choose y axis to be along direction of acceleration.
?Choose x axis to be along the,other” direction of
motion.
? Example,Throwing a baseball (neglecting air resistance).
?Acceleration is constant (gravity).
?Choose y axis up,ay = -g.
?Choose x axis along the ground in the direction of the
throw.
Physics 121,Lecture 6,Pg 10
“x” and,y” components of motion are
independent.
? A man on a train tosses a ball straight up in the air.
?View this from two reference frames:
Reference frame
on the ground.
Reference frame
on the moving train.
y motion,a = -g y
x motion,x = v0t
Physics 121,Lecture 6,Pg 11
Projectile Motion.
? If I set something moving near the earth,it reduces to a 2-D
problem we call projectile motion,
? Use a coordinate system with x along the ground,y vertical
with respect to the ground,(Notice no change in third
direction.)
? Equations of motion reduce to:
?X,?x = voxt ax = 0
?Y,y = yo + voyt – g t2 /2 y positive upwards
Physics 121,Lecture 6,Pg 12
Lecture 6,ACT 2
2-D Motion
? Alice and Bill are playing air hockey on a table with no
bumpers at the ends,Alice scores a goal and the puck goes
flying off the end of the table,Which diagram best describes
the path of the puck?
Alice Bill
A) B) C)
Physics 121,Lecture 6,Pg 13
Problem:
? Sammy Sosa clobbers a fastball toward center-field,You
are checking out your new fancy radar gun which can detect
ball velocity,i.e,speed and direction,You measure that the
ball comes off the bat with initial velocity is 36.5 m/s at an
angle of 30o above horizontal,Since Sammy was hitting a
high fastball,you estimate that he contacted the ball about
one meter off of the ground,You know the dimensions of
Wrigley field and the center-field wall is 371 feet (113m) from
the plate and is 10 feet (3m) high,You decide to
demonstrate your superfast math and physics skills by
predicting whether Sammy get a home run before the play is
decided,
Physics 121,Lecture 6,Pg 14
Problem:
1) We need to find how high the ball is at a distance of 113m
away from where it starts,
?
v
h
D
yo
Physics 121,Lecture 6,Pg 15
Problem:
2) This is a problem in projectile motion.
Choose y axis up.
Choose x axis along the ground in the direction of the hit.
Choose the origin (0,0) to be at the plate.
Say that the ball is hit at t = 0,x = xo = 0,y = yo = 1m
?
v
h
D
y
x
Physics 121,Lecture 6,Pg 16
Problem...
? Variables
?vo = 36.5 m/s
?yo = 1 m
?h = 3 m
??o = 30o
?D = 113 m
?a = (0,ay) ? ay = -g
?t = unknown,
?Yf – height of ball when x=113m,unknown,
our target
Physics 121,Lecture 6,Pg 17
Problem...
3) For projectile motion,
? Equations of motion are:
vx = v0x vy = v0y - g t
x = vx t y = y0 + v0y t - 1/ 2 g t2
And,use geometry to find vox and voy
y
x
g
?
v
v0x
v0yy
0
Find v0x = |v| cos ?.
and v0y = |v| sin ?.
Physics 121,Lecture 6,Pg 18
Problem...4) Solve the problem,
? The time to reach the wall is,t = D / vx (easy!)
? Height at any time,y(t) = y0 + v0y t - g t2/ 2
? Combining the two gives,y(t) = y0 + v0y D/vox - g D2/ (2vox2)
? And substitute for vox and voy,y(t) = y0 + D tan? - g D2/ 2(vocos?)2
? All are known quantities,Solved.
? Numbers:
? y(t) = (1.0 m) + (113 m)(tan 30) -
(0.5)(9.8 m/s2)(113 m)2/(36.5 m/s cos 30)2
= (1.0 + 65.2 - 62.6) m = 3.6 m
Physics 121,Lecture 6,Pg 19
Problem...
5)Think about the answer,
? The units work out correctly for a height (m)
? It seems reasonable for the ball to be a little over 3m
high when it gets to the fence,
? But,we haven’t yet answered the question
? Since the wall is 3m high,and the ball is 3.26m high
when it gets there,Sammy gets a homer.
Physics 121,Lecture 6,Pg 20
? Two footballs are thrown from the same point on
a flat field,Both are thrown at an angle of 30o
above the horizontal,Ball 2 has twice the initial
speed of ball 1.
If ball 1 is caught a distance D1 from the thrower,
how far away from the thrower D2 will the
receiver of ball 2 be when he catches it?
(a) D2 = 2D1 (b) D2 = 4D1 (c) D2 = 8D1
Lecture 6,ACT 3
Motion in 2-D
Physics 121,Lecture 6,Pg 21
Shooting the Monkey
(tranquilizer gun)
? Where does the zookeeper
aim if he wants to hit the monkey?
( He knows the monkey will
let go as soon as he shoots ! )
Physics 121,Lecture 6,Pg 22
Shooting the Monkey...
? If there were no gravity,simply aim
at the monkey
r = r0
r =v0t
Physics 121,Lecture 6,Pg 23
Shooting the Monkey...
r = v0 t - 1/2 g t2
? With gravity,still aim at the monkey! r = r0 -
1/2 g t2
Dart hits the
monkey!
Physics 121,Lecture 6,Pg 24
Recap:
Shooting the monkey...
x = x0
y = -1/2 g t2
? This may be easier to think about,
It’s exactly the same idea!!
x = v0 t
y = -1/2 g t2
Physics 121,Lecture 6,Pg 25
Typical questions,
(projectile motion; for given v0 and ?)
? What is the maximum height the ball reaches (h)?
? How long does it take to reach maximum height?
h = (v0 sin ?) t - 1/2 g t2
v = (v0 sin ?) - g t = 0 at P
t = (v0 sin ?) / g
t = (v0 sin ?) / g !
y,
?
h
L
y
x
v0
P
? Would the answers above be any different if the
projectile was moving only along y-axis (1-D motion)
with the initial velocity,v0 sin (?)?
( A ) YES ( B ) NO ( C ) CAN’T TELL
h
y
xv0 sin(?)
P
Physics 121,Lecture 6,Pg 26
Typical questions,
(projectile motion; for given v0 and ?)
? What is the range of the ball (L)?
? How long does it take for ball to reach final point (P)?
?
h
L
y
x
v0
P
y = (v0 sin ?) t - 1/2 g t2 = 0 ! when at P
[ (v0 sin ?) - 1/2 g t] t = 0
t = 0 ; t = 2 (v0 sin ?) / g
L = vx0 t = (v0 cos ?) tx,
y,
Physics 121,Lecture 6,Pg 27
Problem 2
? Suppose a projectile is aimed at a target at rest placed at
the same height,At the time that the projectile leaves the
cannon the target is released from rest and starts falling
toward ground,Would the projectile miss or hit the target?
t = t1
y
xv0
t = 0
t = 0
TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Physics 121,Lecture 6,Pg 28
Problem 3
? Suppose a projectile is aimed at a target at rest somewhere
above the ground as shown in Fig,below,At the same time
that the projectile leaves the cannon the target falls toward
ground,Would the projectile miss or hit the target?
t = t1
?
y
x
v0
t = 0
t = 0
TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Physics 121,Lecture 6,Pg 29
Inertial Reference Frames:
? A Reference Frame is the place you measure from.
?It’s where you nail down your (x,y,z) axes!
? An Inertial Reference Frame (IRF) is one that is not
accelerating.
?We will consider only IRF’s in this course.
? Valid IRF’s can have fixed velocities with respect to each other,
?More about this later when we discuss forces.
?For now,just remember that we can make measurements
from different vantage points.
Physics 121,Lecture 6,Pg 30
Lecture 6,ACT 4
Relative Motion
? Consider an airplane flying on a windy day.
? A pilot wants to fly from New Haven to Bradley airport,
Having asked a friendly physics student,she knows that
Bradley is 120 miles due north of New Haven and there is
a wind blowing due east at 30 mph,She takes off from
New Haven Airport at noon,Her plane has a compass and
an air-speed indicator to help her navigate,She uses her
compass at the start to aim her plane north,and her air
speed indicator tells her she is traveling at 120 mph with
respect to the air,
After one hour,
A) She is at Bradley
B) She is due east of Bradley
C) She is southeast of Bradley
Physics 121,Lecture 6,Pg 31
Lecture 6,ACT 5
Relative Motion
? You are swimming across a 50m wide river in which the current
moves at 1 m/s with respect to the shore,Your swimming speed is
2 m/s with respect to the water,
You swim across in such a way that your path is a straight
perpendicular line across the river.
?How many seconds does it take you to get across?
2m/s
1m/s50m
s25250 ?a)
s50150 ?b)
s29350 ?c)
s35250 ?d)
Physics 121,Lecture 6,Pg 32
Recap of today’s lecture
? Homework 2,Due Friday Sept,16 @ 6:00PM
?Ch.3,# 2,11,18,20,25,32,36,46,50,and 56.
? Chapter 4,Motion in 2-D
?Review of vectors
?Projectile motion
?Relative velocity
