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
r
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 ///
+=