The Ultimate AP Physics C Mechanics Review
The year is almost over and here is what you need to know for the Mechanics portion of the final and the AP
Physics C Mechanics exam. Hey, it’s only 5 pages; enjoy!
I. One Dimensional Motion
A. Equations of Motion: Assuming a = constant
Be able to use (algebraically) these equations;
Δx = v0Δt + ½ aΔt2
vf = v0 + aΔt
where Displacement = x
Velocity = v
2
2
2
Δx = vfΔt - ½ aΔt
vf = v0 + 2aΔx
Acceleration = a
Δx = ½ (v0 + vf) Δt
B. Graphing:
Understand the relationships between the three graphs of motion;
x vs t (area under v vs t graph)
v vs t (slope of x vs t, or area under a vs t graph)
a vs t (slope of v vs t)
C. Understand the difference between average velocity and instantaneous velocity.
vave = Δx/Δt
vinst = dx/dt which is the slope of the tangent to the x vs t graph at a point.
D. Similarly:
aave = Δv/Δt
ainst = dv/dt which is the slope of the tangent to the v vs t graph at a point.
II. Acceleration due to Gravity
A. Acceleration due to gravity
g = ag = 9.8 m/s2, as measured in our labs.
For objects in free-fall (with no air resistance) g = -9.8 m/s2 if the upward direction is chosen to
be positive. This means that every second an object gains 9.8 m/s of speed if it is moving downward,
or loses 9.8 m/s of speed if it is moving upward. What is the acceleration of an object tossed straight up
at its highest point?
B. Vectors
1. Definition: something that has magnitude and direction
2. Vector addition: the head to tail method
3. Vector subtraction: vf-v0 = vf + (-v0)
C. Motion in two dimensions on Earth’s surface
1. Equations of motion in x and y
2. Projectile Motion: Independence of motions that are at right angles to each other.
The acceleration of projectile is g, straight down, independent of its horizontal motion.
Range Eq: x =(vo2/g) sin 2 where the object lands at the same height it was fired from
(yf = 0)
Projectile Eq: y = tanx – (gx2)/(2vo2cos2) parabolic motion, yf  y0
3. Graphing the parabolic paths of objects in y vs x
D. Introduction to relative motion
1. Moving observer vs “stationary” observer: No preferred reference frame. It is impossible to
say who is “really” moving
2. An observer can feel acceleration but not velocity.
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III. Newton’s Laws and Other Force Stuff
A. Newton’s Laws
Newton’s 1st Law (Swiped from Galileo, but really just a special case of the 2nd Law):
If FNET = 0, then a = 0.
(v = constant)
If a = 0, then FNET = 0.
Newton’s 2nd Law (Physics bedrock, arguably the most important equation in science):
FNET = m a
FNET
Vector relationship!
Newton’s 3rd Law (Important but tricky, even Newton himself made
mistakes applying this law: review the Lawn Mower problem):
Fon A by B = -Fon B by A
F on A by B
A
F on B by A
B
Note: Third Law forces NEVER EVER EVER EVER cancel because they act on two different
objects!
B. Hooke’s Law for springs (Named after Robert (Bob) Hooke, Newton’s sworn enemy):
Spring constant
F = -k x
Force exerted by spring is in opposite direction from displacement.
Many things besides springs obey this law (molecular bonds, compression
waves etc.)
C. Difference between Mass and Weight:
Mass m is universal: same everywhere. Units: kg
Weight depends on the local force of gravity. Units: N
True Weight = Fgrav = mg
Apparent Weight = FN (what the scale reads.)
“Weightlessness” occurs when FN is 0 (free fall: a = g)
D. Friction
1. Kinetic friction fk = k FN
2. Static friction fs max = s FN
Example: skidding, sliding
Example: rolling, turning, no relative motion
between surfaces
IV. Circular Motion and Rotation
Uniform Circular Motion
1. centripetal acceleration ac = v2/R
2. centripetal force
Fc = m ac
Don’t put on FBD! Any force that points toward the center of a circle can be called a
centripetal force: the centripetal force is not an additional force!
3. v = (2  R)/ T where T is the period, or time for one complete circle
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V. Work and Energy
A. Work: W = F.d = |F| |d| cos 
where θ is is the angle between F and d when placed tail to tail
Note W = area under F vs d graph
Or W = ∫ F dx
F
Can be + or -. Unit: Joule J = Nm
W=AREA
F
θ
d
B. Potential Energy “stored ability to do work”
d
1. Ugrav = mgh
(on earth’s surface)
2
2. Uspring = ½ kx
3. F = - dU/dx (the force is the negative slope of the tangent to the U vs. t graph.)
C. Kinetic Energy: K = ½ mv2
(Always +, or 0)
D. Conservation of Energy
1. The Work/Kinetic Energy Theorem: Wtot = K always true!
2. If there is no funny business (Wnc = 0) then mechanical energy E is conserved:
E0 = EF
In other words: Ko + U0 = KF + UF
Conservative forces do no work for any round
trip! You can associate potential energy U with
conservative forces.
E. Three Step problem solving program
1. Draw picture: Label initial and final points, and ref line for gravitational potential
energy.
2. If Wnc = 0 Set Eo = Ef , otherwise use Wnc = E
3. Solve for missing variable
F. Pave = W/t: Power is the rate at which work is done. Units: Watt W=J/s
Pinst = F v (Pinst = Fv cos θ if F and v are not collinear)
VI.
Center of Mass and Momentum
A. For systems of particles, xcm = (1/M) Σmixi
For extended objects, xcm = (1/M) ∫x dm
B. Momentum and Impulse ideas come from Newt’s 2nd Law (where else?)
F = ma
= mΔv/Δt
So FΔt = Δ(mv) , define Momentum p = mv
Then the Impulse FΔt = Δp
Graphically, the Impulse is the area under the F vs t graph
F
Or
p = ∫ F dt
AREA =p
Can be + or -. Unit: N∙s = kg∙m/s
t
Note that we can also write F = Δp/Δt
Which means the instantaneous force F = dp/dt
Which means you can get the force by finding the slope of the p vs t graph
C. Impulse Applications
Safety : crash helmets, freeway barricades, airbags etc.
The idea is to minimize F by maximizing t
3
t
D. Conservation of Momentum:p0 = pf if Fext Δt= 0
m1v10 + m2v2o = m1v1f + m2v2f
p1f
p2f
E. Collisions in 1 and 2 dimensions always conserve momentum
1. if elastic, then conserve K too.
2. if inelastic: don’t conserve K
VII. Rotation
A. Equations of Motion
θ = ω0Δt + ½ αΔt2
θ = ωfΔt - ½ αΔt2
p1
0
Is momentum
conserved in this
collision? How
would you test
this graphically?
p2
Assuming α = constant
0
ω = ω0 + αΔt
where Angular Displacement = θ
Angular Velocity = ω
2
2
ωf = ω0 + 2αθ
Angular Acceleration = α
θ = ½ (ω0 + ωf) Δt
B. Connection between linear and angular variables
θ = s/r
ω = v/r
α = aT/r
(ar = ω2r)
r
θ
s
C. Rotational Inertia I:
For discrete masses: I = Σmr2
For continuous masses: I = ∫r2 dm
Parallel Axis Theorem I = Icm + Mh2
dm
r
D. Torque
Def: τ = r x F = |r| |F| sin 
τ=Iα
θ
r
F
τ
E. Energy
If τext = 0, then Eo = Ef. When doing energy problems you need to account for both the
translational and rotational kinetic energy: K = ½ mv2 + ½ Icmω2
Work (Kinetic) Energy Theorem W = ΔK
Work W = τ θ
Power P = τ ω
r
F. Angular Momentum
For a discrete mass: l = r x p
For an extended mass: L = I ω
Note that τ = dL/dt
l
θ
p
l=rxp
Conservation of Angular Momentum: if τext Δt = 0, Lo = Lf
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VIII. Simple Harmonic Motion SHM
A. Def of SHM: anytime the restoring force is directly proportional to displacement x from
equilibrium: Frestore = -kx
B. Reference Circle: If you have SHM, then the projection of uniform circular motion onto a
diameter will represent the motion. It is much easier to analyze uniform circular motion than linear
motion with nonconstant acceleration.
The motion equation for SHM is x = A sin(ωt), where ω is the angular frequency: ω = 2π/T
Where T is the period (the time it takes for one complete cycle.)
Note that T = 1/f, where f is the frequency.
C. If SHM, then the period T of oscillation will be independent of the amplitude of the oscillation and
you can use the reference circle to show that
T  2 m
k
D. Applications:
i. Mass on a spring (see equation for T above)
ii. Pendulum T  2 l g
iii. Motion through a hole in the Earth (T = 84 minutes, remember?)
VIII. Universal Law of Gravity and Orbits
A. Kepler’s Laws
1st Law: Planets travel in elliptical orbits with sun at one of the foci
2nd Law: Planets sweep out equal area in equal time
3rd Law: T2/Rave3 = Constant
B. Newton’s Universal Law of Gravity
Rave
mm
Fgrav  G 1 2 2
r
C. Orbits: Elliptical orbits are equivalent to circular orbits
where we let the radius of the circle r be equal to the semi-major
axis (Rave) of the ellipse
D. Set Fg = mac to find:
1. Mass of orbited body (orbiting mass always drops out)
2. Period T of orbit
3. Radius of orbit (including geosynchronous orbits)
4. Speed of orbiting object
E. Universal Gravitational Potential Energy
U grav  G
r
Fg
m1m2
r
F. Escape Speed: apply conservation of energy E0=Ef, set Ef = 0, vesc = SQRT(2GM/r)
G. Black holes: set the escape speed equal to the speed of light c.
H. Tides caused by differential gravitational pulls.
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r