Kinematics of 3- or 2-dimensional motion

1
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
2
Equations of 3-D Kinematics for Constant
Acceleration
Result 3-D motion with constant acceleration is
a superposition of three independent motions
along x, y, and z axes.
3
Projectile Motion
ax0 ? vxv0xconst ay -g ? vy voy- gt x x0
vox t y yo voy t gt2/2 v0x v0 cos a0
v0y v0 sin a0 tan a vy / vx
Exam Example 6 Baseball Projectile
Data v022m/s, a040o
(examples 3.7-3.8, problems 3.12)
Find (a) Maximum height h (b) Time of flight
T (c) Horizontal range R (d) Velocity when
ball hits the ground
Solution
v0x22m/scos40o17m/s v0y22m/ssin40o14m/s

• vy0 ? h (vy2-v0y2) / (2ay) - (14m/s)2 / (-
  2 9.8m/s2) 10 m
• y (v0yvy)t / 2 ? t 2y / v0y 2 10m /
  14m/s 1.45 s T 2t 2.9 s
• R x v0x T 17 m/s 2.9 s 49 m
• vx v0x , vy - v0y

4
Motion in a Circle

• Uniform circular motion
• v const

Centripetal acceleration
Magnitude ac v2 / r Direction to center
(b) Non-uniform circular motion v ? const
5
Exam Example 7 Ferris Wheel (problems 3.29)
Data R14 m, v0 3 m/s, a 0.5 m/s2

• Find
• Centripetal acceleration
• Total acceleration vector
• Time of one revolution T

Solution
(a) Magnitude ac a- v2 / r Direction to
center
?
(b)
(c)
6
Relative Velocity
c

Flying in a crosswind
Correcting for a crosswind
7
Principles of Special Theory of Relativity
(Einstein 1905)

• Laws of Nature are invariant for all inertial
  frames of reference.
• (Mikelson-Morlys 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 Galileos
ones
y
y
Lorentz transformation
x
V
x
Proved by Fizeau experiment (1851) of light
dragging by water
Contraction of length
Slowing down of time
Twin paradox
Slowing and stopping light in gases (predicted at
Texas AM)