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Introductory Mechanics - Notes on Equations | PHYS 250, Study notes of Physics

Material Type: Notes; Class: INTRODUCTORY MECHANICS; Subject: Physics (Univ); University: Western Kentucky University; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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koofers-user-9rz 🇺🇸

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Phys 250, Equations
Vectors
A
B
B
A
R
+
=
+
=
)( BABA
+=
θ
cosAAx=
θ
sinAAy=
22
yx AAA +=
=
x
y
A
A
1
tan
θ
jAiAA yx ˆˆ +=
jRiRjBAiBABAR yxyyxx ˆˆˆ
)(
ˆ
)( +=+++=+=
Motion in one dimension
12 xxx =
t
x
tt
xx
vxav
=
=
12
12
dt
dx
t
x
vt
x=
=
0
lim
t
v
tt
vv
axxx
xav
=
=
12
12
dt
dv
t
v
axx
t
x=
=
0
lim
2
2
dt
xd
dt
dx
dt
d
dt
dv
ax
x=
==
tavv xxx += 0
2
00
2
1tatvxx xx ++=
)(2 0
2
0
2xxavv xxx +=
t
vv
xx xx
+
= 2
0
0
Circular motion
v
arad
2
=,
dt
vd
a
=
tan
2
2
4
T
R
arad
π
=
Motion in two or three
dimensions
kziyixr ˆ
ˆˆ ++=
t
r
tt
rr
vav
=
=
12
12
dt
rd
t
r
vt
=
=
0
lim
k
dt
dz
j
dt
dy
i
dt
dx
dt
rd
vˆ
ˆˆ ++==
t
v
tt
vv
aav
=
=
12
12
dt
vd
t
v
at
=
=
0
lim
k
dt
dv
j
dt
dv
i
dt
dv
dt
vd
az
y
xˆ
ˆˆ ++==
2
2
dt
xd
dt
dv
ax
x== 2
2
dt
yd
dt
dv
ay
y==
2
2
dt
zd
dt
dv
az
z==
k
dt
zd
j
dt
yd
i
dt
xd
aˆ
ˆˆ 2
2
2
2
2
2
++=
Projectile motion
(
)
tvx = 00 cos
α
( )
2
00
2
1
sin tgtvy =
α
00 cos
α
= vvx
tgvvy= 00 sin
α
( )
2
0
22
0
0cos2
tan x
v
g
xy
=
α
α
g
v
h
=2
sin 0
22
0
α
g
v
R0
2
02sin
α
=
Relative Velocity
ABBPAP vvv ///
+=

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Phys 250, Equations

Vectors

R A B B A

A B A ( B )

Ax =Acos θ

Ay =Asin θ

2 2 A = Ax +A y

x

y

A

A

1

θ tan

A =Axi ˆ^ +Ayˆ j

R = A+B=( Ax +Bx)iˆ+(Ay+By)jˆ=Rxiˆ+Ry jˆ

Motion in one dimension

∆x =x 2 −x 1

t

x

t t

x x v (^) av x ∆

−^ =

2 1

2 1

dt

dx

t

x v t

x = ∆

∆ → 0

lim

t

v

t t

v v a

x x x av x ∆

−^ =

2 1

2 1

dt

dv

t

v a

x x

t

x = ∆

∆ → 0

lim

2

2

dt

d x

dt

dx

dt

d

dt

dv a

x x = 

v (^) x = v 0 x+ax⋅ t

2 0 0 2

x =x +vx ⋅t+ ⋅ax⋅t

2 0

2 v (^) x =vx+ ⋅ax⋅ x−x

t

v v x x

x x ⋅ 

0 0

Circular motion

r

v arad

2

= , dt

d v a

tan^ =

2

2 4

T

R

arad

Motion in two or three

dimensions

r xi yi z k

t

r

t t

r r vav ∆

2 1

2 1

dt

d r

t

r v t

∆ → 0

lim

k dt

dz j dt

dy i dt

dx

dt

d r v

t

v

t t

v v aav ∆

2 1

2 1

dt

d v

t

v a t

∆ → 0

lim

k dt

dv j dt

dv i dt

dv

dt

d v a

x (^) ˆ yˆ zˆ = = + +

2

2

dt

d x

dt

dv a

x x =^ = 2

2

dt

d y

dt

dv a

y y = =

2

2

dt

d z

dt

dv a

z z = =

k dt

d z j dt

d y i dt

d x a ˆ^ ˆ ˆ 2

2

2

2

2

2

= + +

Projectile motion

x = ( v 0 ⋅cos α 0 ) ⋅t

2 0 0 2

y =v ⋅sin α ⋅t− ⋅g⋅t

vx= v 0 ⋅cos α 0

vy = v 0 ⋅sin α 0 −g⋅ t

2

0

2 2 0

0 2 cos

tan x v

g y x 

g

v h ⋅

sin (^0)

2 2

g

v R

0

2

0 ⋅sin^2 α

Relative Velocity

vP (^) /A vP/B vB/ A