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AP Physics C: Mechanics
2015 Free-Response Questions
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ADVANCED PLACEMENT PHYSICS C TABLE OF INFORMATION
CONSTANTS AND CONVERSION FACTORS
Proton mass, m p
1.67 – 10 27 kg
Neutron mass, mn
1.67 – 10 27 kg
Electron mass, me
9.11 – 10 31 kg
6.02 – 10 23 mol 1
Avogadro’s number, N 0
8.31 J (mol <K)
R
Universal gas constant,
1.60 – 10 19 C
1 electron volt, 1 eV
1.60 – 10 19 J
Speed of light,
Universal gravitational
constant,
Acceleration due to gravity
at Earth’s surface,
3.00 – 108 m s
c
6.67 – 10 11 N<m 2
G
kg 2
9.8 m s2
g
1.38 – 10 23 J K
Boltzmann’s constant, k B
1 unified atomic mass unit,
1u
1.66 – 10 27 kg
931 MeV c 2
Planck’s constant,
h
6.63 – 10 34 J <s
4.14 – 10 15 eV <s
hc
1.99 – 10 25 J <m
1.24 – 103 eV < nm
e0
8.85 – 10 12 C2 N < m 2
Vacuum permittivity,
1 4 pe0 Coulomb’s law constant, k
Vacuum permeability,
m0
m0 4 p Magnetic constant, k „
1 atmosphere pressure,
UNIT
SYMBOLS
e
Electron charge magnitude,
meter,
kilogram,
second,
ampere,
kelvin,
PREFIXES
Factor
Prefix
Symbol
m
kg
s
A
K
mole,
hertz,
newton,
pascal,
joule,
1 atm
mol
Hz
N
Pa
J
9.0 – 109 N< m 2
C2
4 p – 10 7 (T <m) A
1 – 10 7 (T< m) A
1.0 – 105 N m 2
watt,
coulomb,
volt,
ohm,
henry,
W
C
V
W
H
1.0 – 105 Pa
farad,
tesla,
degree Celsius,
electron volt,
F
T
’C
eV
VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
D
q
0
30D
37D
45D
53D
60D
90D
109
giga
G
sin q
0
12
35
2 2
45
3 2
1
106
mega
M
cos q
1
3 2
45
2 2
35
12
0
103
kilo
k
tan q
0
3 3
34
1
43
3
‡
10 2
centi
c
10 3
milli
m
6
micro
m
10 9
nano
n
10 12
pico
p
10
The following assumptions are used in this exam.
I. The frame of reference of any problem is inertial unless otherwise
stated.
II. The direction of current is the direction in which positive charges
would drift.
III. The electric potential is zero at an infinite distance from an isolated
point charge.
IV. All batteries and meters are ideal unless otherwise stated.
V. Edge effects for the electric field of a parallel plate capacitor are
negligible unless otherwise stated.
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ADVANCED PLACEMENT PHYSICS C EQUATIONS
MECHANICS
Ãx
Ãx 0 a x t
x
x0 Ãx 0 t Ãx2
a
1 2
at
2 x
Ãx 0 2 2 a x x x0 ÇF
Fnet
m
m
F
dp
dt
J
Ô F dt
p
mv
Dp
F f … m FN
Ô F dr
DE
W
K
1 2
mÃ
2
P
dE
dt
P
F v
DUg
ac
t
a
I
mg Dh
Ã2
r
w r
r–F
Çt
I
t net
I
Ô r dm
2
Ç mr
Ã
rw
L
r – p
Iw
1 2
Iw
2
w
w0 at
q
q0 w0 t acceleration
energy
force
frequency
height
rotational inertia
impulse
kinetic energy
spring constant
length
angular momentum
mass
power
momentum
radius or distance
period
time
potential energy
velocity or speed
work done on a system
position
coefficient of friction
angle
torque
angular speed
angular acceleration
phase angle
T
1
f
Ts
2p
Tp
2p
FG
m
k
DV
Ô E dr
r2
Gm1m2
r
dV
dx
V
1
4pe0
UE
qV
DV
Q
C
Cp
Ç Ci
ÇC
i
1
=
=
=
=
=
=
=
V=
v =
r =
F =
k =
P
Q
q
R
r
t
U
ÔB
dQ
dt
1
QDV
2
area
magnetic field
capacitance
distance
electric field
emf
force
current
current density
inductance
length
number of loops of wire
per unit length
number of charge carriers
per unit volume
power
charge
point charge
resistance
radius or distance
time
potential or stored energy
electric potential
velocity or speed
resistivity
flux
dielectric constant
qv – B
FM
i
i
=
=
=
=
=
=
=
=
=
=
=
=
N =
1 q1q2
4pe0 r
k e0 A
d
UC
n
q
Ç rii
i
C
1
C DV 2
2
dB
d
m0 I
m0 I d – r
4p r 2
R
r
A
F
ÔI d
E
rJ
Bs
m0 nI
I
Nevd A
FB
ÔB
I
DV
R
e
ÔE
e
L
g
Gm1m2
F
I
J
L
I
2p
w
Q
e0
Ex
1
2
k Dx 2
A
B
C
d
E
ε
dA
Us
xmax cos( wt f
FE
q
ÔE
1
Cs
UG
1 2
at
2
E
k D x
x
1 q1q2
4pe0 r 2
FE
Fs
2
Ç mi xi
Ç mi
xcm
K
2
a =
E=
F =
f =
h =
I =
J =
K =
k =
=
L =
m=
P =
p =
r =
T =
t =
U=
v =
W=
x =
m =
q =
t =
w =
a =
f =
ELECTRICITY AND MAGNETISM
Rs
1
Rp
P
-3-
Ç Ri
i
1
ÇR
i
I DV
i
UL
–B
dA
d
dI
dt
1 2
LI
2
d FB
dt
ADVANCED PLACEMENT PHYSICS C EQUATIONS
GEOMETRY AND TRIGONOMETRY
Rectangle
A bh
A = area
C = circumference
V = volume
S = surface area
b = base
h = height
= length
w = width
r = radius
s = arc length
q = angle
Triangle
A
1
bh
2
Circle
A
pr2
C
2p r
s rq
Rectangular Solid
V
wh
Cylinder
V
S
pr
CALCULUS
s
2
df
dx
S
c2
b
c
tan q
a
b
a cos ax d
>cos ax @
dx
a sin ax n
dx
ax
dx
1 n 1
x , n › 1
n 1
1 ax
e
a
ln x a
Ô cos ax dx
1
sin ax a
Ô sin ax dx
1
cos ax a
VECTOR PRODUCTS
a
c
cos q
1
x
d
>sin ax @
dx
dx
Right Triangle
sin q
aeax
Ôxa
2
a 2 b2
d ax
e dx
Ôe
2
4 3
pr
3
4p r
nx n 1
Ôx
r
Sphere
V
d n
x dx
d
ln ax dx
q
2p r 2p r
d f du
du dx
AB
c
q
a
A–B
90°
b
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AB cos q
AB sin q
2015 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time— 45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this
booklet in the spaces provided after each part.
Mech.1.
A block of mass m is projected up from the bottom of an inclined ramp with an initial velocity of magnitude v0 .
The ramp has negligible friction and makes an angle q with the horizontal. A motion sensor aimed down the
ramp is mounted at the top of the incline so that the positive direction is down the ramp. The block starts a
distance D from the motion sensor, as shown above. The block slides partway up the ramp, stops before reaching
the sensor, and then slides back down.
(a) Consider the motion of the block at some time t after it has been projected up the ramp. Express your
answers in terms of m, D, v0 , t, q and physical constants, as appropriate.
i. Determine the acceleration a of the block.
ii. Determine an expression for the velocity v of the block.
iii. Determine an expression for the position x of the block.
(b) Derive an expression for the position x min of the block when it is closest to the motion sensor. Express your
answer in terms of m, D, v0 , q , and physical constants, as appropriate.
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2015 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(c) On the axes provided below, sketch graphs of position x, velocity v, and acceleration a as functions of
time t for the motion of the block while it goes up and back down the ramp. Explicitly label any intercepts,
asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
(d) After the block slides back down and leaves the bottom of the ramp, it slides on a horizontal surface with a
coefficient of friction given by mk . Derive an expression for the distance the block slides before stopping.
Express your answer in terms of m, D, v0 , q , mk , and physical constants, as appropriate.
(e) Suppose the ramp now has friction. The same block is projected up with the same initial speed v0 and
comes back down the ramp. On the axes provided below, sketch a graph of the velocity v as a function of
time t for the motion of the block while it goes up and back down the ramp, arriving at the bottom of the
ramp at time t f . Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or
algebraic expressions, as appropriate.
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2015 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech.2.
A small dart of mass 0.020 kg is launched at an angle of 30° above the horizontal with an initial speed of 10 m/s.
At the moment it reaches the highest point in its path and is moving horizontally, it collides with and sticks to a
wooden block of mass 0.10 kg that is suspended at the end of a massless string. The center of mass of the block
is 1.2 m below the pivot point of the string. The block and dart then swing up until the string makes an angle q
with the vertical, as shown above. Air resistance is negligible.
(a) Determine the speed of the dart just before it strikes the block.
(b) Calculate the horizontal distance d between the launching point of the dart and a point on the floor directly
below the block.
(c) Calculate the speed of the block just after the dart strikes.
(d) Calculate the angle q through which the dart and block on the string will rise before coming momentarily to
rest.
(e) The block then continues to swing as a simple pendulum. Calculate the time between when the dart collides
with the block and when the block first returns to its original position.
(f) In a second experiment, a dart with more mass is launched at the same speed and angle. The dart collides
with and sticks to the same wooden block.
i. Would the angle q that the dart and block swing to increase, decrease, or stay the same?
_____ Increase
_____ Decrease
_____ Stay the same
Justify your answer.
ii. Would the period of oscillation after the collision increase, decrease, or stay the same?
_____ Increase
_____ Decrease
_____ Stay the same
Justify your answer.
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2015 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech.3.
A uniform, thin rod of length L and mass M is allowed to pivot about its end, as shown in the figure above.
(a) Using integral calculus, derive the rotational inertia for the rod around its end to show that it is ML2 3 .
The rod is fixed at one end and allowed to fall from the horizontal position A through the vertical position B.
(b) Derive an expression for the velocity of the free end of the rod at position B. Express your answer in terms
of M, L, and physical constants, as appropriate.
An experiment is designed to test the validity of the expression found in part (b). A student uses rods of
various lengths that all have a uniform mass distribution. The student releases each of the rods from the
horizontal position A and uses photogates to measure the velocity of the free end at position B. The data are
recorded below.
Length (m)
0.25
0.50
0.75
1.00
1.25
1.50
Velocity (m/s)
2.7
3.8
4.6
5.2
5.8
6.3
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2015 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(c) Indicate below which quantities should be graphed to yield a straight line whose slope could be used to
calculate a numerical value for the acceleration due to gravity g.
Horizontal axis:
Vertical axis:
Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not
given. Label each row you use and include units.
(d) Plot the straight line data points on the grid below. Clearly scale and label all axes, including units as
appropriate. Draw a straight line that best represents the data.
(e)
i. Using your straight line, determine an experimental value for g.
ii. Describe two ways in which the effects of air resistance could be reduced.
STOP
END OF EXAM
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