Chapter Opener

1
Relative Velocity Fixed and Moving Frames

• v turtles velocity with respect to moving
  frame (ruler)
• V rulers velocity with respect to fixed
  frame (lab)
• v turtles velocity with respect to lab

V
Hecht (Physics Calculus)

• we see that v v V
• same idea in 2d and 3d v v V

2
An Ant Ambulates at an Angle

• V papers velocity
• v ants velocity on the paper
• v v V ants actual velocity with respect
  to the earth

V
Hecht (Physics Calculus)
v
v
3
An Airplane Aspires to Actually Aim eAst

• V winds velocity
• v airplanes velocity in the wind
• v v V airplanes actual velocity with
  respect to the earth

v
v
V
v
4

• Example A kayaker desires to cross a river
  directly to the opposite shore. The kakak moves
  at 2 m/s. The river flows at .5 m/s and it is 50
  m wide.
• Find the direction to paddle in (b) find the time

Answer Let x be the direction of flow, and y be
directly across the river. The canoe is pointed
at an angle q upstream from y. V .5 i and v
v j where v is unknown. And, v 2 j cos q
2 i sin q. Since v v V, we get x
direction 0 - 2 sin q .5 ? sin q .5/2 ? q
14.5º y direction v 2 cos q 1.94 m/s So
the time is t distance/speed 50 m/(1.94 m/s)
25.8 s
5
The Average Acceleration Vector a
The Acceleration Vector a
Take limit of a as Dt ? 0 so we get

• Its magnitude is the instantaneous acceleration
  a
• Its direction is not at all obviously related to
  the path

6
In General, v Will Change Direction and Magnitude
If a is Parallel or Antiparallel to v, the
DIRECTION of v does not change although it could
possibly flip direction
7
Two Classic Simple 2d Motions
This anticipates PROJECTILE VOMITING
This anticipates PROJECTILE MOTION (a constant
down)
This anticipates UNIFORM CIRCULAR MOTION (a
constant direction toward a point)
8
The First Simple MotionProjectile Motion
Kinematics

• a constant - j g in the xy plane
• vo i v0 cos q 0 j v0 sin q 0
• or vx0 v0 cos q 0 and vy0 v0 sin q 0
• along x and y we have velocity components
• vx vx0 and vy vy0 gt
• and for displacement components we get
• x x0 vx0 t and y y0 vy0 t - 1/2
  gt2
• STEADY MOTION along x FREE-CLIMB/FALL along y
• take initial values x0 y0 0
• eliminate t from the displacement equations
  since t x/vx0

A quadratic function y(x) GRAVITYS RAINBOW!!
9
Projectile motion is uniformly accelerated

• plane of motion defined by plane of a and v0
• angle of v0 is launch angle q0
• speed of launch is v0
• all flights are parabolas in space!!
• motion is constant velocity along x and pure
  free-fall along y

Maximum Height
10
Combining the x and y Motions Projectile Motion
11
Interesting Times Range on Level Ground Maximum
Height

• time-to-peak tP occurs when vy 0 so vy0 g tP
  ? tP vy0 /g
• for flight on level ground, at landing y 0, so
  total flight time tF

• range is flights total horizontal distance
• for level ground put tF into x equation

• maximum height occurs at time tP

12
A graph of range vs launch angle, for 4
different initial speeds
13
A Lovely Symmetry in the Range Maximum Range

• can show that ON LEVEL GROUND the range for a
  launch angle q0 is the same as the range for a
  launch angle 90º - q0
• can show that ON LEVEL GROUND the maximum range
  occurs for q0 45º

14
Example
To find where hammer lands, solve the quadratic
equation for x
15
The Other Simple Motion Uniform Circular Motion
Kinematics

• in projectile motion a vector constant
  (simple)
• uniform acceleration ? parabolic motion (messy!)
• what is simplest non-boring motion?
• a circle of constant radius r about a center
  point, at constant speed v uniform circular
  motion
• acceleration direction toward the center
  centripetal acceleration also called radial
  acceleration
• acceleration magnitude constant a v2/r
• in fact, any curved motion can be approximated
  by a circle of some radius, perhaps changing from
  point to point. If speed changes too, a has a
  centripetal (perpendicular) AND a tangential
  (parallel) component

16
The Displacement Triangle is Similar to the
Velocity Triangle

• displacement q arclength/r ? v Dt /r
• velocity q ? a Dt /v
• triangles are geometrically SIMILAR
• equate and cancel Dt ? a v2/r
• if speed is constantfor small Dt, Dv ? v, so a
  is also perpendicular to v
• a is centripetal

17

• Informative Example How often should the earth
  rotate so that the acceleration of gravity is the
  centripetal acceleration, at the equator?
• (If this is the case, objects at the equator
  would be weightless)

• answer g v2/r, with v surface velocity of
  the earth at equator
• since v distance/time 2pr/T (r earths
  radius T earths rotation period) we get g
  4p2r/T2 or T2 4p2r/g
• T2 (4p2)(6.4 x 106 m)/(9.8 m/s2) 2.6 x 107
  s2
• so T 5100 s 1.4 hours
• orbital period of low-earth-orbit objects like
  space shuttle
• useful notion here since a 4p2r/T2
• define angular frequency w 2p/T 2p
  radians/period
• ? a w2r