Physics 121,Lecture 7,Pg 1
Physics 121,Sections 9,10,11,and 12
Lecture 7
Today’s Topics:
? Homework 3,Due Friday Sept,23 @ 6:00PM
?Ch.3,# 64,75,and 81.
?Ch.4,# 4,8,21,25,36,and 40.
? Chapter 4,Motion in 2-D
?Projectile motion
?Relative velocity
?More examples of FBD’s
Physics 121,Lecture 7,Pg 2
Typical questions,
(projectile motion; for given v0 and q)
? What is the maximum height the ball reaches (h)?
? How long does it take to reach maximum height?
h = (v0 sin q) t - 1/2 g t2
v = (v0 sin q) - g t = 0 at P
t = (v0 sin q) / g
t = (v0 sin q) / g !
y,
q
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 (q)?
( A ) YES ( B ) NO ( C ) CAN’T TELL
h
y
xv0 sin(q)
P
Physics 121,Lecture 7,Pg 3
Typical questions,
(projectile motion; for given v0 and q)
? What is the range of the ball (L)?
? How long does it take for ball to reach final point (P)?
q
h
L
y
x
v0
P
y = (v0 sin q) t - 1/2 g t2 = 0 ! when at P
[ (v0 sin q) - 1/2 g t] t = 0
t = 0 ; t = 2 (v0 sin q) / g
L = vx0 t = (v0 cos q) tx,
y,
Physics 121,Lecture 7,Pg 4
Problem 1
? 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 7,Pg 5
Problem 2
? 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
q
y
x
v0
t = 0
t = 0
TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Physics 121,Lecture 7,Pg 6
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 7,Pg 7
Lecture 7,ACT 1
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 7,Pg 8
Lecture 7,ACT 2
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 7,Pg 9
Physics 121,Lecture 7,Pg 10
Dynamics
? Isaac Newton (1642 - 1727) published Principia Mathematica
in 1687,In this work,he proposed three,laws” of motion:
Law 1,An object subject to no external forces is at rest or moves
with a constant velocity if viewed from an inertial reference
frame.
Law 2,For any object,FNET = ??F = ma
Law 3,Forces occur in pairs,FA,B = - FB,A
(For every action there is an equal and opposite reaction.)
Physics 121,Lecture 7,Pg 11
Newton’s First Law
An object subject to no external forces moves with a constant
velocity if viewed from an inertial reference frame.
?If no forces act,there is no acceleration.
? The above statement can be thought of as the definition of inertial
reference frames.
?An IRF is a reference frame that is not accelerating (or
rotating) with respect to the,fixed stars”.
?If one IRF exists,infinitely many exist since they are related by
any arbitrary constant velocity vector!
Physics 121,Lecture 7,Pg 12
Is Storrs a good IRF?
? Is Storrs accelerating?
? YES!
?Storrs is on the Earth.
?The Earth is rotating,
? What is the centripetal acceleration of Storrs?
?T = 1 day = 8.64 x 104 sec,
?R ~ RE = 6.4 x 106 meters,
? Plug this in,aS =,034 m/s2 ( ~ 1/300 g)
? Close enough to 0 that we will ignore it.
? Storrs is a pretty good IRF.
RT2RRva
22
S ?????????
?? 2
Physics 121,Lecture 7,Pg 13
Newton’s Second Law
The acceleration of an object is directly proportional to the net force
acting upon it,The constant of proportionality is the mass.
amFF NE T ??? ???
? Units
?The units of force are kg m/s2 = Newtons (N)
?The English unit of force is Pounds (lbs)
Physics 121,Lecture 7,Pg 14
Lecture 7,ACT 3
Newton’s Second Law
A constant force is exerted on a cart that is initially at rest on
an air table,The force acts for a short period of time and
gives the cart a certain final speed.
Air Track
CartForce
For a second shot,I can apply a force only half as large (I’m
getting tired),To reach the same final speed,for how long
must I apply the force?
A) 4 x as long B) 2 x as long C) Same time
D) 1/2 as long E) 1/4 x as long
Physics 121,Lecture 7,Pg 15
Lecture 7,ACT 4
Newton’s Second Law
I push with a force of 2 Newtons on a cart that is initially at rest
on an air table with no air,I push for a second,Because
there is no air,the cart stops after I finish pushing,It has
traveled a certain distance,
Air Track
CartForce
For a second shot,I push just as hard but keep pushing for 2
seconds,The distance the cart moves the second time
versus the first is,
A) 4 x as long B) 2 x as long C) Same
D) 1/2 as long E) 1/4 x as long
Physics 121,Lecture 7,Pg 16
Lecture 7,ACT 5
Newton’s Second Law
A constant force is exerted on a cart that is initially at rest on an air
table,This force is applied for a short period of time and the cart
acquires a certain final speed,which I call vf1.
Air Track
Cart Force
I repeat the experiment,but this time the cart is already moving
with constant speed when I start applying the force,After
exerting the same constant force for the same time interval,
the cart’s final speed is,
A) vf1 B) 2vf1 C) vf12
D) cannot be determined from the information given.
Physics 121,Lecture 7,Pg 17
Newton’s Third Law:
If object 1 exerts a force on object 2 (F2,1 ) then object 2
exerts an equal and opposite force on object 1 (F1,2)
F1,2 = -F2,1
This is among the most abused concepts in physics.
REMEMBER,
Newton’s 3rd law concerns force pairs which
act on two different objects (not on the same object) !
For every,action” there is an equal and opposite,reaction”
Physics 121,Lecture 7,Pg 18
An Example
FB,E = - mB g
EARTH
FE,B = mB g
Consider the forces on an object undergoing
projectile motion
FB,E = - mB g
FE,B = mB g
Physics 121,Lecture 7,Pg 19
Lecture 7,ACT 6
Newton’s Third Law
A fly gets smushed onto the windshield of a speeding bus.
?
The force exerted by the bus on the fly is,
A) greater than
B) the same as
C) less than
that exerted by the fly on the bus.
Physics 121,Lecture 7,Pg 20
Lecture 7,ACT 7
Newton’s Third Law
A fly gets smushed onto the windshield of a speeding bus.
?
The acceleration due to this collision of the bus is,
A) greater than
B) the same as
C) less than
that of the fly.
Physics 121,Lecture 7,Pg 21
Newton's Third Law...
? FA,B = - FB,A an example,
Fm,w Fw,m
Fm,f
Ff,m
Physics 121,Lecture 7,Pg 22
Example of Bad Thinking
? Since Fm,b = -Fb,m why isn’t Fnet = 0,and a = 0?
a
Fm,b Fb,m
ice
Physics 121,Lecture 7,Pg 23
Example of Good Thinking
? Consider only the box as the system!
?Free Body Diagram
Fm,b Fb,m
ice
Physics 121,Lecture 7,Pg 24
Example of Good Thinking
? Consider only the box as the system!
?Free Body Diagram
abox
Fb,m
FgFN
abox = Fb,m/mbox
Physics 121,Lecture 7,Pg 25
Free Body Diagram
A heavy sign is hung between two poles by a rope at each
corner extending to the poles.
Eat at Bob’s
What are the forces on the sign?
Physics 121,Lecture 7,Pg 26
Free Body Diagram
Eat at Bob’s
q1 q2
T1 T2
Add vectors
q2
mg
q1
T1 T2mg
Physics 121,Lecture 7,Pg 27
Free Body Diagram
Eat at Bob’s
q1 q2
T1 T2
mg
Vertical,
mg = T1sinq1 + T2sinq2
Horizontal,
T1cosq1 = T2cosq2
Physics 121,Lecture 7,Pg 28
Normal Forces and String Tension
Certain forces act to keep an object in place,
These have what ever force needed to balance all others
(until a breaking point).
FT,B
FB,T
Physics 121,Lecture 7,Pg 29
Force Pairs
Newton’s 3rd law concerns force pairs,Two members of
a force pair cannot act on the same object,
Don’t confuse gravity (the force of the earth on an object)
and normal forces,It’s an extra part of the problem,
FT,B
FB,T
FB,E = -mg
FE,B = mg
Physics 121,Lecture 7,Pg 30
Lecture 7,ACT 8
Newton’s 3rd Law
? Two blocks are being pushed by a finger on a horizontal frictionless
floor,How many action-reaction pairs of forces are present in this
system?
(a) 2 (b) 4 (c) 6
a b
Physics 121,Lecture 7,Pg 31
An Example
Consider the following two cases
Physics 121,Lecture 7,Pg 32
An Example
The Free Body Diagrams
mg
mg
FB,T= N
Ball Falls For Static Situation
N = mg
Physics 121,Lecture 7,Pg 33
An Example
The action/reaction pair forces
FB,E = -mg F
B,T= N
FE,B = mg
FB,E = -mg
FE,B = mg
FT,B= -N
Physics 121,Lecture 7,Pg 34
Lecture 7,ACT 9
Gravity and Normal Forces
A woman in an elevator is accelerating upwards
The normal force exerted by the elevator on the woman is,
A) greater than
B) the same as
C) less than
the force due to gravity acting on the woman
Physics 121,Lecture 7,Pg 35
Lecture 7,ACT 9b
Gravity and Normal Forces
A woman in an elevator is accelerating upwards
The normal force exerted by the elevator on the woman is,
A) greater than
B) the same as
C) less than
the force the woman exerts on the elevator.
Physics 121,Lecture 7,Pg 36
Exercise,Inclined plane
? A block of mass m slides down a frictionless ramp
that makes angle q with respect to horizontal,
What is its acceleration a?
q
m a
Physics 121,Lecture 7,Pg 37
Inclined plane...
? Define convenient axes parallel and
perpendicular to plane:
?Acceleration a is in x direction only.
q
m a
i
j
Physics 121,Lecture 7,Pg 38
Inclined plane...
? Consider x and y components separately:
? i,mg sin q = ma a = g sin q
? j,N - mg cos q??0,N = mg cos q
mg
N
mg sin q
mg cos q
q
ma
i
j
Physics 121,Lecture 7,Pg 39
Angles of an Inclined plane
q
q
f
ma = mg sin q
qq+?f???90?
mg
N
Physics 121,Lecture 7,Pg 40
Recap of today’s lecture
? Homework 3,Due Friday Sept,23 @ 6:00PM
?Ch.3,# 64,75,and 81.
?Ch.4,# 4,8,21,25,36,and 40.
? Chapter 4,Motion in 2-D
?Projectile motion
?Relative velocity
?More examples of FBD’s,etc.
Physics 121,Sections 9,10,11,and 12
Lecture 7
Today’s Topics:
? Homework 3,Due Friday Sept,23 @ 6:00PM
?Ch.3,# 64,75,and 81.
?Ch.4,# 4,8,21,25,36,and 40.
? Chapter 4,Motion in 2-D
?Projectile motion
?Relative velocity
?More examples of FBD’s
Physics 121,Lecture 7,Pg 2
Typical questions,
(projectile motion; for given v0 and q)
? What is the maximum height the ball reaches (h)?
? How long does it take to reach maximum height?
h = (v0 sin q) t - 1/2 g t2
v = (v0 sin q) - g t = 0 at P
t = (v0 sin q) / g
t = (v0 sin q) / g !
y,
q
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 (q)?
( A ) YES ( B ) NO ( C ) CAN’T TELL
h
y
xv0 sin(q)
P
Physics 121,Lecture 7,Pg 3
Typical questions,
(projectile motion; for given v0 and q)
? What is the range of the ball (L)?
? How long does it take for ball to reach final point (P)?
q
h
L
y
x
v0
P
y = (v0 sin q) t - 1/2 g t2 = 0 ! when at P
[ (v0 sin q) - 1/2 g t] t = 0
t = 0 ; t = 2 (v0 sin q) / g
L = vx0 t = (v0 cos q) tx,
y,
Physics 121,Lecture 7,Pg 4
Problem 1
? 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 7,Pg 5
Problem 2
? 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
q
y
x
v0
t = 0
t = 0
TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Physics 121,Lecture 7,Pg 6
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 7,Pg 7
Lecture 7,ACT 1
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 7,Pg 8
Lecture 7,ACT 2
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 7,Pg 9
Physics 121,Lecture 7,Pg 10
Dynamics
? Isaac Newton (1642 - 1727) published Principia Mathematica
in 1687,In this work,he proposed three,laws” of motion:
Law 1,An object subject to no external forces is at rest or moves
with a constant velocity if viewed from an inertial reference
frame.
Law 2,For any object,FNET = ??F = ma
Law 3,Forces occur in pairs,FA,B = - FB,A
(For every action there is an equal and opposite reaction.)
Physics 121,Lecture 7,Pg 11
Newton’s First Law
An object subject to no external forces moves with a constant
velocity if viewed from an inertial reference frame.
?If no forces act,there is no acceleration.
? The above statement can be thought of as the definition of inertial
reference frames.
?An IRF is a reference frame that is not accelerating (or
rotating) with respect to the,fixed stars”.
?If one IRF exists,infinitely many exist since they are related by
any arbitrary constant velocity vector!
Physics 121,Lecture 7,Pg 12
Is Storrs a good IRF?
? Is Storrs accelerating?
? YES!
?Storrs is on the Earth.
?The Earth is rotating,
? What is the centripetal acceleration of Storrs?
?T = 1 day = 8.64 x 104 sec,
?R ~ RE = 6.4 x 106 meters,
? Plug this in,aS =,034 m/s2 ( ~ 1/300 g)
? Close enough to 0 that we will ignore it.
? Storrs is a pretty good IRF.
RT2RRva
22
S ?????????
?? 2
Physics 121,Lecture 7,Pg 13
Newton’s Second Law
The acceleration of an object is directly proportional to the net force
acting upon it,The constant of proportionality is the mass.
amFF NE T ??? ???
? Units
?The units of force are kg m/s2 = Newtons (N)
?The English unit of force is Pounds (lbs)
Physics 121,Lecture 7,Pg 14
Lecture 7,ACT 3
Newton’s Second Law
A constant force is exerted on a cart that is initially at rest on
an air table,The force acts for a short period of time and
gives the cart a certain final speed.
Air Track
CartForce
For a second shot,I can apply a force only half as large (I’m
getting tired),To reach the same final speed,for how long
must I apply the force?
A) 4 x as long B) 2 x as long C) Same time
D) 1/2 as long E) 1/4 x as long
Physics 121,Lecture 7,Pg 15
Lecture 7,ACT 4
Newton’s Second Law
I push with a force of 2 Newtons on a cart that is initially at rest
on an air table with no air,I push for a second,Because
there is no air,the cart stops after I finish pushing,It has
traveled a certain distance,
Air Track
CartForce
For a second shot,I push just as hard but keep pushing for 2
seconds,The distance the cart moves the second time
versus the first is,
A) 4 x as long B) 2 x as long C) Same
D) 1/2 as long E) 1/4 x as long
Physics 121,Lecture 7,Pg 16
Lecture 7,ACT 5
Newton’s Second Law
A constant force is exerted on a cart that is initially at rest on an air
table,This force is applied for a short period of time and the cart
acquires a certain final speed,which I call vf1.
Air Track
Cart Force
I repeat the experiment,but this time the cart is already moving
with constant speed when I start applying the force,After
exerting the same constant force for the same time interval,
the cart’s final speed is,
A) vf1 B) 2vf1 C) vf12
D) cannot be determined from the information given.
Physics 121,Lecture 7,Pg 17
Newton’s Third Law:
If object 1 exerts a force on object 2 (F2,1 ) then object 2
exerts an equal and opposite force on object 1 (F1,2)
F1,2 = -F2,1
This is among the most abused concepts in physics.
REMEMBER,
Newton’s 3rd law concerns force pairs which
act on two different objects (not on the same object) !
For every,action” there is an equal and opposite,reaction”
Physics 121,Lecture 7,Pg 18
An Example
FB,E = - mB g
EARTH
FE,B = mB g
Consider the forces on an object undergoing
projectile motion
FB,E = - mB g
FE,B = mB g
Physics 121,Lecture 7,Pg 19
Lecture 7,ACT 6
Newton’s Third Law
A fly gets smushed onto the windshield of a speeding bus.
?
The force exerted by the bus on the fly is,
A) greater than
B) the same as
C) less than
that exerted by the fly on the bus.
Physics 121,Lecture 7,Pg 20
Lecture 7,ACT 7
Newton’s Third Law
A fly gets smushed onto the windshield of a speeding bus.
?
The acceleration due to this collision of the bus is,
A) greater than
B) the same as
C) less than
that of the fly.
Physics 121,Lecture 7,Pg 21
Newton's Third Law...
? FA,B = - FB,A an example,
Fm,w Fw,m
Fm,f
Ff,m
Physics 121,Lecture 7,Pg 22
Example of Bad Thinking
? Since Fm,b = -Fb,m why isn’t Fnet = 0,and a = 0?
a
Fm,b Fb,m
ice
Physics 121,Lecture 7,Pg 23
Example of Good Thinking
? Consider only the box as the system!
?Free Body Diagram
Fm,b Fb,m
ice
Physics 121,Lecture 7,Pg 24
Example of Good Thinking
? Consider only the box as the system!
?Free Body Diagram
abox
Fb,m
FgFN
abox = Fb,m/mbox
Physics 121,Lecture 7,Pg 25
Free Body Diagram
A heavy sign is hung between two poles by a rope at each
corner extending to the poles.
Eat at Bob’s
What are the forces on the sign?
Physics 121,Lecture 7,Pg 26
Free Body Diagram
Eat at Bob’s
q1 q2
T1 T2
Add vectors
q2
mg
q1
T1 T2mg
Physics 121,Lecture 7,Pg 27
Free Body Diagram
Eat at Bob’s
q1 q2
T1 T2
mg
Vertical,
mg = T1sinq1 + T2sinq2
Horizontal,
T1cosq1 = T2cosq2
Physics 121,Lecture 7,Pg 28
Normal Forces and String Tension
Certain forces act to keep an object in place,
These have what ever force needed to balance all others
(until a breaking point).
FT,B
FB,T
Physics 121,Lecture 7,Pg 29
Force Pairs
Newton’s 3rd law concerns force pairs,Two members of
a force pair cannot act on the same object,
Don’t confuse gravity (the force of the earth on an object)
and normal forces,It’s an extra part of the problem,
FT,B
FB,T
FB,E = -mg
FE,B = mg
Physics 121,Lecture 7,Pg 30
Lecture 7,ACT 8
Newton’s 3rd Law
? Two blocks are being pushed by a finger on a horizontal frictionless
floor,How many action-reaction pairs of forces are present in this
system?
(a) 2 (b) 4 (c) 6
a b
Physics 121,Lecture 7,Pg 31
An Example
Consider the following two cases
Physics 121,Lecture 7,Pg 32
An Example
The Free Body Diagrams
mg
mg
FB,T= N
Ball Falls For Static Situation
N = mg
Physics 121,Lecture 7,Pg 33
An Example
The action/reaction pair forces
FB,E = -mg F
B,T= N
FE,B = mg
FB,E = -mg
FE,B = mg
FT,B= -N
Physics 121,Lecture 7,Pg 34
Lecture 7,ACT 9
Gravity and Normal Forces
A woman in an elevator is accelerating upwards
The normal force exerted by the elevator on the woman is,
A) greater than
B) the same as
C) less than
the force due to gravity acting on the woman
Physics 121,Lecture 7,Pg 35
Lecture 7,ACT 9b
Gravity and Normal Forces
A woman in an elevator is accelerating upwards
The normal force exerted by the elevator on the woman is,
A) greater than
B) the same as
C) less than
the force the woman exerts on the elevator.
Physics 121,Lecture 7,Pg 36
Exercise,Inclined plane
? A block of mass m slides down a frictionless ramp
that makes angle q with respect to horizontal,
What is its acceleration a?
q
m a
Physics 121,Lecture 7,Pg 37
Inclined plane...
? Define convenient axes parallel and
perpendicular to plane:
?Acceleration a is in x direction only.
q
m a
i
j
Physics 121,Lecture 7,Pg 38
Inclined plane...
? Consider x and y components separately:
? i,mg sin q = ma a = g sin q
? j,N - mg cos q??0,N = mg cos q
mg
N
mg sin q
mg cos q
q
ma
i
j
Physics 121,Lecture 7,Pg 39
Angles of an Inclined plane
q
q
f
ma = mg sin q
qq+?f???90?
mg
N
Physics 121,Lecture 7,Pg 40
Recap of today’s lecture
? Homework 3,Due Friday Sept,23 @ 6:00PM
?Ch.3,# 64,75,and 81.
?Ch.4,# 4,8,21,25,36,and 40.
? Chapter 4,Motion in 2-D
?Projectile motion
?Relative velocity
?More examples of FBD’s,etc.
