PHYSICS: MIDTERM MATHEMATICAL MODELS
NAME:
CHAPTER I – INTRODUCTION TO PHYSICS
PHYSICAL
QUANTITY
SYMBOL
UNITS
MATH MODEL
Distance
d
m
-
How far an object is located from the origin.
Scalar
Distance + direction
Vector
Change between two positions
Vector
m/s
Change between two positions over change in time.
Scalar
m/s
Speed + direction
Slope of position vs. clock reading graph.
Vector
Change between two velocities over change in time.
Slope of velocity vs. clock reading graph.
Vector
Position
dx
or
dy
m
-
Displacement
dx
or
dy
m
dx = dxf - dxi
TYPE OF PHYSICAL
QUANTITY
DEFINITION
CHAPTER II – MOTION WITH CONSTANT VELOCITY
Speed
Velocity
v
vx
or
vy
CHAPTER III – Motion with constant acceleration
Acceleration
ax
or
ay
m/s2
vxf - vxi
a
t
CHAPTER II & CHAPTER III – MOTION WITH CONSTANT VELOCITY & MOTION WITH CONSTANT ACCELERATION
Position
MWCV
MWCA
dxf  dxi  vx  t 
1

dxf  dxi  vx  t     ax  t 2 
2

vxf  vxi  ax  t 
Velocity
(constant)
acceleration
ax  0
m
s2
Displacement
(Area under the graph)
dx  vx  t
Velocity after any
displacement
(constant)
vxf - vxi
a
t
(constant)
dx 
1
 vf  vi  t
2
vxf 2  vxi2  2  ax  dx 
CHAPTER IV – VECTORS
Vertical component (y-component)
By B sin

Vectors: Physical quantities that require of a magnitude (number) and a direction.

Examples: Position, displacement, velocity, acceleration, Force, Impulse,
Momentum, Electric current, magnetic field, Electric Field.
Bx = B · cos (  )

Scalars: Physical quantities that require only of a magnitude (number)
Resultant

Examples: Mass, volume, density, temperature, speed, distance, energy, work,
power, Voltage, Resistance.
By = B · sin (  )
Horizontal component (x-component)
Pythagorean Theorem
B2 = ( Bx)2 + (By)2
CHAPTER V – FORCES
Force of Earth
FEonO
N
Force that Earth exerts on an object.
Vector
Mass
m
kg
Amount of matter in an object.
scalar
Gravitational
constant
g
m/s2
Acceleration of an object due to the force exerted by Earth
Slope of FEonO vs. mass.
Vector
Force of the Spring
FSonO
N
Force that a spring exerts on an object.
Vector
Elongation
L
m
Change between two lengths of a stretched/compressed spring.
Vector
Spring constant
k
N/m
Ratio between the force exerted by the spring on an object and the
elongation caused on the spring.
Slope of FSonO vs. Elongation.
scalar
Force of kinetic
Friction
Fk
N
Force exerted by a surface, parallel to the surface.
Vector
Normal Force
FN
N
Force exerted by a surface, perpendicular to the surface.
Vector
Coefficient of kinetic
Friction
k
-
Ratio between two force Fk & FN
Slope of Fk vs. FN.
scalar
Tension Force
FT
N
Force exerted by a cable, rope, thread, string, etc.
vector
FSonO
k
L
Fk
k FN
-
CHAPTER VI – PROJECTILE MOTION
FREE FALL
VERTICAL LAUNCH
HORIZONTAL LAUNCH
ANGLED LAUNCH
Position
HOW FAR
(at any given t)
dy = ( ½ · g · t2)
dy = ( ½ · g · t2) + ( vyi · t ) + (dyi)
dy = ( ½ · g · t2)
dy = ( ½ · g · t2) + ( vyi · t ) + (dyi)
Velocity
HOW FAST
(at any given t)
Vy = ( g · t) + vyi
Vy = ( g · t) + vyi
Vy = ( g · t) + vyi
Vy = ( g · t) + vyi
Acceleration
g = -9.8
(at any given t)
Time to reach
maximum
height
g = -9.8
t TOP =
N/A
GOING UP
Time in Flight
COMING
DOWN
m
s2
t =
2 · dy
vyi  0
Initial velocity
m
s
Initial velocity
Maximum
height
2 · dy
g
vyi = vyTOP - (g · tTOP)
( vyi )2 = ( vyTOP )2 - (2 · g · dyTOP)
N/A
(NO t)
g = -9.8
vy TOP - vyi
g
t =
g
m
s2
dy TOP =
N/A
(vy TOP )2 - (vy i )2
(NO t)
2·g
m
s2
t TOP =
N/A
t =
2 · dy
g
vyi  0
N/A
N/A
g = -9.8
m
s
t =
m
s2
vy TOP - vyi
g
2 · dy
g
vyi = vyTOP - (g · tTOP)
( vyi )2 = ( vyTOP )2 - (2 · g · dyTOP)
dy TOP =
(vy TOP )2 - (vy i )2
2·g
CHAPTER VII – CIRCULAR MOTION
Circumference
c
m
Tangential velocity
vT
m/s
Centripetal
acceleration
aC
m/s2
Centripetal Force
FC
N
c  2r
 r
T
v2
ac =
r
v2
Fc  m
r
vT =
2
Perimeter of a circle
Scalar
Velocity of an object moving in circular motion. Tangential to the
circle and perpendicular to the Centripetal Force.
Vector
Acceleration of an object moving in circular motion. Towards the
center of the circle.
Vector
Any type Force that keeps an object moving in circular motion.
Towards the center of the circle.
Vector
PHYSICS: FINAL EXAM MATHEMATICAL MODELS
NAME:
CHAPTER VIII – UNIVERSAL GRAVITATION
Gravitational Force
FG
N
Gravitational
constant
g
m2 / s2
Universal
gravitational
constant
G
N · m2 /
kg2
Velocity of an
orbiting object
ving
m/s
FG = G
m1
m2
2
r
m
g = G 2P
rP
G = 6.673 x 10-11
G · morbited
rcircle
Ving =
Force of attraction between two objects of masses m1 and m2.
Vector
Gravitational constant of any planet.
Vector
Constant of proportionality
Velocity of an object orbiting another object.
Vector
CHAPTER IX – IMPULSE & MOMENTUM
Momentum
p
kgm/s
Impulse
I
Ns
p
m v
I
F
t
p
CONSERVATION OF MOMENTUM
i

p
Vector
Product of Force times change in clock reading.
Force exerted over a period of time that changes the momentum
of an object.
Vector
The initial momentum of system can change only if an impulse is delivered on the system.
The final momentum of the system will equal the initial momentum of the system plus the
delivered impulse on the system.
Pi + I = pF
IMPULSE CHANGES MOMENTUM
Amount of motion of an object.
f
p1i  p 2i  I  p1F  p 2F
The total momenta of the system are conserved after a collision.
The total momenta before the collision equal the total momenta after the collision.
CHAPTER X – WORK – POWER - ENERGY
Work
W
J
W = F · d
Force parallel to d
Gravitational
Potential Energy
UG
J
Kinetic Energy
KE
Elastic Potential
Energy
Power
Potential to do something
Scalar
UG = g · m · dy
Potential to do something due to a change in a vertical position
(dy)
Scalar
J
KE = ½ · m · v2
Potential to do something due to a change in a velocity (v)
Scalar
US
J
US = ½ · k · L2
Potenti Potential to do something due to a change in the
elongation (L) of an elastic object.
Scalar
P
W
P = W / t
Rate at which work is done
scalar
UGi + W = UGF
KEi + W = KEF
USi + W = USF
WORK CHANGES ENERGY
CONSERVATION OF MECHANICAL ENERGY
The initial energy of the system can change only if work is done on the system. The final
energy of the system will equal the initial energy of the system plus the work done on the
system.
The total mechanical energy of the system is neither created nor destroyed.
UGi + KEi + USi + W = UGF
+ KEF + USF
The total mechanical energy is conserved.
v=
2KE
m
CHAPTER XI – TORQUE

Torque
Property of a force to make an object rotate.
N·m
When the force causing the torque: FEonO = g · m

F r
ROTATIONAL EQUILIBRIUM
Force perpendicular to r
CW
Vector
   CCW
The sum of the Torques in the counter clockwise direction (ccw) equal the sum of the
Torques in the clockwise direction (cw)
CHAPTER XII – ELECTRICITY
Electrostatics
constant
Electric force
Coulomb’s law
K
N · m2 /
C2
Fq1onq2
N
Electric field
(at any point)
Electric field
(felt by a charged
particle)
K = 9 x 109
Fq1onq2 = K
E=K
E
N/m
E=
Electric Potential
Energy
Uq
J
Uq = K
Electric potential
V
v
V=
v
v
Uq
J
Change in electric
potential
Constant of proportionality
q1
q2
2
d
q
d2
Attractive or repulsive force between two charges of charge q1
and q2
Vector
Space around an electric Charge. Electric field felt at any point
around that electric charge.
Vector
Fq
Space around an electric Charge. Electric field felt by a charge
place around that electric charge.
q
q1
q2
Potential to change the velocity of charged particle.
scalar
Potential to change the energy of an electric charge placed
between two electric plates with a constant Electric field (E) and a
distance (d).
scalar
V = E · d
Difference between two different electric potentials.
scalar
Uq = V · q
Energy that changes when a charge moves from one place to
another place.
scalar
d
(K · qA )
d
(Voltage)
Change in Electric
Potential Energy
CHAPTER XIII – ELECTRIC CIRCUITS
Electric current
Ohm’s law
I
A
I
A
V
V
R
q
I
Flow of electric charge (q) per unit of time (t).
t
V
I R

Vector
Charges in motion
Electric current (I) is directly proportional to the change in electric
potential (V).
Vector
Electric current (I) is inversely proportional to the resistance (R).
P  I  V
Power
(Electric)
P
P  I2  R
W
Rate at which electric work is done.
2

V 
P
Scalar
R
PHYSICAL QUANTITY
CIRCUITS IN SERIES
CIRCUITS IN PARALLEL
Stays the same
IT = I1 + I2 + I3 +. . .
VT = V1 + V2 + V3 +. . .
Stays the same
RT = R 1 + R 2 + R 3 + . . .
1
1
1
1
=
+
+
+ ...
RT
R1 R 2
R3
PT = P1 + P2 + P3 +. . .
PT = P1 + P2 + P3 +. . .
ELECTRIC CURRENT
VOLTAGE
RESISTANCE
POWER
CHAPTER XIV – MAGNETISM
Magnetic force on a
charged particle
Magnetic force on a
current carrying
wire
FB
FB
N
N
FB = B
FB = L
v
B
q
I
sin 
Force (FB) felt by a charged particle (q) that travels with velocity
(v) perpendicular to a magnetic field (B). The direction of this
force is perpendicular to both the direction of the magnetic field
and the direction of the velocity of the particle.
Vector
sin 
Force (FB) felt by a wire of length (L) carrying an electric current
(I) placed perpendicular to a magnetic field (B). The direction of
this magnetic force is perpendicular to both the direction of the
magnetic field and the direction of the electric current.
Vector
CHAPTER XIV – WAVES
Period of a simple pendulum:
T =2·  ·
T=
L =
G=
L
g
Period
Length of the string
gravitational constant
(seconds)
(meters)
(m/s2)
Period of a mass-spring system:
T =2·  ·
T=
m =
k=
m
k
Period
mass
Spring constant
Period
T
s
Frequency
f
Hz
Wavelength

m
Speed (wave)
v
m/s
(seconds)
(kilograms)
(N/m)
1
T f
v
f
v T


Time for one cycle.
Distance peak to peak measured in the “time graph”.
Scalar

Number of cycles in one second.
Scalar


Length of one cycle.
Distance peak to peak measured in the “position graph”.
Vector
Change in position over the change in time of a disturbance.
Fundamental Frequency of a standing wave on a string or in an open pipe
f1 =
v
( 2·L )
f1 =
v =
L=
Fundamental frequency
wave speed
Length of spring or open pipe
(Hertz)
(kilograms)
(meters)
Frequency of a any harmonic of a standing wave on a string or in an open pipe
fn = n ·
v
( 2·L )
(n = 1, 2, 3, 4, . . . )
fn =
v =
L=
n=
Frequency of the harmonic
wave speed
Length of spring or open pipe
number of the harmonic
(Hertz)
(kilograms)
(meters)
Vector