matics of 3- or 2-dimensional moti z x y v 1 v 2 v Position vector: z y x i z i y i x r Average velocity: t r t t r r v 1 2 1 2 Instantaneous velocity: z y x i dt dz i dt dy i dt dx dt r d t r v lim Average acceleration: t v t t v v a 1 2 1 2 Instantaneo us acceleratio n: 2 2 lim dt r d dt v d t v a 0 t 0 t z y x i dt z d i dt y d i dt x d a 2 2 2 2 2 2 a a a || a || → magnitude of velocity a ┴ → direction of velocity
Kinematics of 3- or 2-dimensional motion. z. Position vector:. Average velocity:. Instantaneous velocity:. y. x. Average acceleration:. Instantaneous acceleration:. a || → magnitude of velocity a ┴ → direction of velocity. Equations of 3-D Kinematics for Constant Acceleration. - PowerPoint PPT Presentation
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Kinematics of 3- or 2-dimensional motionz
x y
v
1v
2v
Position vector: zyx iziyixr
Average velocity:t
r
tt
rrv
12
12
Instantaneous velocity:
zyx idt
dzi
dt
dyi
dt
dx
dt
rd
t
rv
lim
Average acceleration:t
v
tt
vva
12
12
Instantaneous acceleration: 2
2
limdt
rd
dt
vd
t
va
0t
0t
zyx idt
zdi
dt
ydi
dt
xda
2
2
2
2
2
2
aaa
||
a|| → magnitude of velocitya┴ → direction of velocity
Equations of 3-D Kinematics for Constant Acceleration
)1( tavv
0
)2( tvv
rr
2
00
)3( 200 2
1tatvrr
)4( )(2 020
2 rravv
Result: 3-D motion with constant acceleration is a superposition of three independent motions along x, y, and z axes.
a
Projectile Motionax=0 → vx=v0x=constay= -g → vy= voy- gtx = x0 + vox ty = yo + voy t – gt2/2v0x= v0 cos α0 v0y= v0 sin α0 tan α = vy / vx Exam Example 6: Baseball Projectile Data: v0=22m/s, α0=40o
x0 y0 v0x v0y ax ay x y vx vy t
0 0 ? ? 0 -9.8m/s2 ? ? ? ? ?Find: (a) Maximum height h;(b) Time of flight T;(c) Horizontal range R; (d) Velocity when ball hits the ground
Principles of Special Theory of Relativity (Einstein
1905):1. Laws of Nature are invariant for all inertial frames of reference. (Mikelson-Morly’s experiment (1887): There is no “ether wind” ! )2. Velocity of light c is the same for all inertial frames and sources.
Relativistic laws for coordinates transformation and addition of velocities are not Galileo’s ones:
Contraction of length:
Slowing down of time: Twin paradox
Slowing and stopping light in gases (predicted at Texas A&M)
y y’
x
x’
V
v
xvyv
22
2
22 /1
/,
/1 cV
cVxtt
cV
tVxx
2
22
2 /1
/1,
/1 cVv
cVvv
cVv
Vvv
x
yy
x
xx
22 /1 cVxx
22 /1 cVtt
Proved by Fizeau experiment (1851)of light dragging by water )/( ncv