Quadratic formula: Kinematics under constant acceleration: Average ...

Average & instantaneous velocity
and acceleration:
~v average =
! x
t
~v = d~x
dt
Kinematics under constant acceleration:
~v (t) =~v (t 0 )+~a [t t 0 ] v x (t) 2 = v x (t 0 ) 2 +2a x [x(t) x(t 0 )]
~r (t) =~r (t 0 )+~v (t 0 )[t t 0 ]+ 1 2 ~a [t t 0] 2 ...and similarly for y and z components.
~a average =
! v
t
~a = d~v
dt = d 2 ~x
dt 2
~r (t) =~r (t 0 )+ 1 2 [~v (t)+~v (t 0)] t
Vectors:
Forces:
F net = ma
⇥A · ⇥B = AB cos
= A x B x + A y B y + A z B z
|F gravity | = mg
F kinetic friction = µ k N
A = |A| = A 2 x + A 2 y + A 2 z A x = A cos A y = A sin
Where θ for components is measured
counter-clockwise from the î direction.
F spring = k(x x 0 ) F static friction µ s N
F gravity = GmM
r 2
(between two point masses)
Uniform circular
motion:
a centripetal = v2
r
v =
r
T = 2 r
v
Quadratic formula:
ax 2 + bx + c =0 x = b ± ⇥ b 2 4ac
2a
Drag Forces:
~F = bv ˆv
~F = 1 2 C⇢Av2 ˆv
(low speed)
(high speed)
(v with a hat on it is a unit vector
in the direction of the velocity).
Work and kinetic
energy (linear and
rotational):
r f
K = 1 W = F · dr
2 mv2 W = K f K i
K = 1 2 I 2 W =
r i
f
i
⇤⇥ · ⇤d

Angular velocity and acceleration:
⇥ = d dt
= d⇥
dt
Angular ↔ Linear analogies:
Center of Mass, Momentum and Impulse:
1D Collision, elastic:
Angular motion at constant angular acceleration:
⇥(t) =⇥(t 0 )+ [t t 0 ]
⇤(t) 2 = ⇤(t 0 ) 2 +2 [⇥(t) ⇥(t 0 )]
⇥(t) =⇥(t 0 )+⇤(t 0 )[t t 0 ]+ 1 2
[t t 0 ] 2
Since these equations describe rotation
x v a m I
Torque and angular momentum:
⇥F ⇥ p L
⇤⇥ = I⇤ = d⇤ L
dt = ⇤r F ⇤ ⇥
1
L = I ⇥ = ⇥r ⇥p
2 mv2 $ 1 2 I!2 Potential Energy & Energy Conservation: F = dU
dx
Z
R center of mass = 1 m i r i = 1 xf
Z ~r f
M total
M
rdm
U = U f U i = Fdx = F ~ · dr ~ = Wcons
x i ~r i
i
~p = m~v
Z
K 1 + U 1 + W non-cons = K 2 + U 2 U gravity = mgy
t2
~J = ~p 2 ~p 1 = F ~ (t)dt
~F U spring = 1 2 k(x x 0) 2 U gravity =
GmM
net = d~p
t 1
dt
r
Power:
P average =
W P = F · v
v f = m 1v 1i + m 2 v 2i
P = dW m 1 + m 2
t
dt P = ⇤ · ⇤⇥
v 1f = m 1 m 2
v 1i + 2m 2
v
The formula for v2f is obtained by swapping 1↔2 in
2i
m 1 + m 2 m 1 + m 2 the formula to the left.
ring or hollow cylinder (about center): mr 2
2
solid sphere (about center):
I = m i ri 2 = r 2 5
dm
mr2
1
i
disc, solid cylinder (about center): 2 mr2 2
hollow sphere (about center): 3 mr2
I = I CoM + MD 2 1
1
thin rod (about center): 12 mL2 thin rod about end: 3 mL2
1D Collision, totally inelastic:
Moment of inertia:
a tangential = ↵r
v tangential = !r
around a fixed axis, they involve the
components of θ, ω and α along this axis.

Equation of motion of mass-spring system:
kx m d2 x
dt 2 = =
General solution:
x(t) =x m cos(!t + )
k
m
T = 1 f = 2 ⇥
Simple pendulum:
E = K + U = 1 2 kx2 m
= g L
Solution for a damped oscillator: x(t) =x = b
m e
t cos(!t + )
2m
Traveling simple harmonic wave: y(x, t) =y m sin(kx !t + )
Period: T = 2 ⇥ = 1 f
k = 2⇥
Wave on string:
Some potentially useful mathematics:
sin 30 =0.5
d
cos 30 ⇡ 0.866
dt (atn )=an t n 1
sin(90 ✓) = cos ✓
Z
at n dt =
a
n +1 tn+1
tan ✓ = sin ✓
cos ✓
v wave = T
= ⇥ k
! =
v wave =
(sin ✓) 2 + (cos ✓) 2 =1
r
k
m
T string
µ
sin(A ± B) =sinA cos B ± cos A sin B
cos(A ± B) = cos A cos B ⌥ sin A sin B
2

Gravitational Constant G : 6.67259 × 10 -11 N m 2 /kg 2
Acceleration due to gravity (g) : 9.80 m/s 2
Average earth-moon distance : 3.84 × 10 8 m
Average earth-sun distance : 1.49 × 10 11 m
Average radius of the earth : 6.37 × 10 6 m
Average radius of Mars : 3.37 × 10 6 m
Mass of the earth : 5.97 × 10 24 kg
Mass of the moon : 7.36 × 10 22 kg
Mass of the sun : 1.99 × 10 30 kg
Mass of Mars : 6.42 × 10 23 kg
Force
1 N = 10 5 dyne
1 N = 0.2248 lb
Power
1 hp = 550 ft lb/s
1 W = 1 J/s
1 W = 0.738 ft lb/s
Time
60 s = 1 min.
60 min. = 1 hr.
Mass
1000 kg = 1 metric ton
1000 g = 1 kg
1 slug = 14.59 kg
Length
1 in. = 2.54 cm
1 ft. = 12 in.
1 yd. = 3 ft.
1 m = 39.37 in.
1 mile = 1.609 km
1 Å = 10 -10 m
1 lightyear = 9.461 × 10 15 m
1 µm = 10 -6 m
1 u = 1.66 × 10 -27 kg
Energy
1 J = 10 7 erg
1 J = 0.738 ft lb
1 cal = 4.186 J
1 Btu = 252 cal
1 eV = 1.6 × 10 -19 J
1 kWh = 3.6 × 10 6 J
931.5 MeV = 1 u