Motion in Two and Three Dimensions

1
Chapter-4

• Motion in Two and Three Dimensions

2
Ch 4-2 Position and Displacement in Two and
Three Dimensions

• Position vector r in three dimension
• r xi yj zk
• x -3 m, y 2m and z5m

3
Ch 4-2 Position and Displacement in Three
Dimensions

• Displacement ?r
• ?r r2-r1
• where r1 x1iy1jz1k
• r2 x2iy2jz2k
• ?r (x2- x1) i (y2- y1) j (z2 z1) k
• ?r (?x) i (?y) j (?z)
• ?x x2- x1 ?y y2- y1 ?z z2- z1

4
Checkpoint 4-1

• ?r (?x)i (?y)j (?z)k
• ?x 6-(-2)8
• ?y -2-4 -6
• ?z -3-(-3)0
• ?r 8i - 6j
• (b) The displacement vector ?r is parallel to xy
  plane

• (a) If a wily bat flies from xyz coordinates (
  -2m, 4 m, -3 m) to coordinates ( 6 m, -2 m, -3
  m), what is its displacement ?r in unit vector
  notation?
• (b) Is ?r paralell to one of the three
  coordinates planes? If so, which one?

5
Ch 4-3 Average and Instantaneous Velocity

• Average velocity vavg
• vavg displacement/time interval
• vavg ?r /?t
• ?r/?t (?x/?t) i (?y/?t) j (?z/?t)k
• Instantaneous velocity v
• v dr/dt
• v (dx/dt) i (dy/dt) j (dz/dt) k
• vvxivyjvzk
• Direction of v always tangent to the particle
  path at the particle position

6
Checkpoint 4-1(new book)

• The figure shows a circular path taken by a
  particle. If the instantaneous velocity of the
  paticle is v (2 m/s) i- (2 m/s) j
• , through which quadrant is the particle moving
  at that instant if it is traveling
• (a) clockwise
• (b) counterclockwise around the circle?

• Ans v (2 m/s) i- (2 m/s) j
• (a) clockwise motion
• vx ve and vy -ve only in
• First quadrant
• (b) counterclockwise motion
• vx ve and vy -ve only in
• Third quadrant

7
Ch 4-4 Average and Instantaneous Acceleration

• Average acceleration aavg
• aavg change in velocity/time
• aavg ?v/?t (v2-v1)/ ?t
• Instantaneous acceleration a
• a dv/dt
• a (dvx/dt)i (dvy/dt)j (dvz/dt)k
• aaxiayjazk

8
Checkpoint 4-2

• (1) d2x/dt2 -6 , d2y/dt212
• (2) d2x/dt2 -18 t ,d2y/dt2-10
• (3) d2r/dt2 4 i ,
• (4) d2r/dt2 24 t i ,
• ax is constant for 1 and 3,ay is constant for
  both ay is so a is constant
• ax is not constant for 2 and 4 while ay is
  constant. Hence a is not constant for case 2 and 4

• Here are four description of the positions of a
  puck as it moves in the XY plane
• (1) X -3t24t-2 and y 6t2 4t
• (2) X -3t3 -4t and Y -5t2 6
• (3) r 2t2 i (4t 3) j
• (4) r (4t3 -2)i 3 j

9
Checkpoint 4-4

• r(4t3-2t)i3j
• 4t3i-2ti3j
• Since units of 4t3-2t and 3 has to be meter then
• Unit of 4 is m/s2
• Unit of -2 is m/s
• Unit of 3 is m

• If the position of a hobos marble is given by
• r(4t3-2t)i3j, with r in meters and t in
  seconds, what must be the units of coefficients
  4, -2 and 3?

10
Ch 4-5 Projectile Motion

• Motion in two dimension
• Horizontal motion (along x-axis ) with constant
  velocity, ax0
• Vertical motion (along y-axis) with constant
  acceleration ay g
• Horizontal motion and the vertical motion
  independent of each other
• v0v0xiv0yj
• v0xv0 cos? v0yv0 sin?

11
Ch 4-5 Projectile Motion

• Horizontal motion
• x-x0v0xt v0cos?t
• Horizontal Range R
• R is the horizontal distance traveled by the
  projectile when it has returned to its initial
  launch height
• R v0 cos? t (v02sin2?)/g
• Rmaxv02/g (?45?)

12
Ch 4-6 Projectile Motion Analyzed

• Vertical motion
• y-y0v0yt gt2/2
• v0 sin?t gt2/2
• vy v0y t gt v0 sin?t gt
• Max. height h v0y2 /2?g ?
• Projectile path equation
• Projectile path is a parabola given by
• y(tan?) x - gx2/2vx2

13
Formule Summary for Projectile Motion
Equation of motion Horizontal Motion (ax0) Vertical Motion (ay g)
vavg (vivf)/2 vavg v0x v0 cos? Vavg (v0yvy)/2 v0y v0 sin?
vf vi at vx v0x v0 cos? vy v0 sin? -gt
(vf)2 (vi)2 2ax (vx)2 (v0 cos?)2 (vy)2 (v0 sin?)2 -2gy
x vitat2/2 x v0 cos? t y v0 sin? t- (gt2)/2
Max. Distance R 2 v0 cos? tup (v0)2 sin2? /2g Rmax (v0)2 /2g h (v0 sin? )2 / 2g
14
Checkpoint 4-4

• A fly ball is hit to the outfield. During its
  flight (ignore the effect of the air), what
  happens to its (a) horizontal and (b)vertical
  components of velocity
• What are its (c) horizontal and (d) vertical
  components of its acceleration during ascent,
  during descent, and at the topmost point of its
  flight?

• vxconstant
• vy initially positive and then decreases to zero
  and finally it increases in negative value.
• ax 0
• ay -g
• throughout the entire projectile path

15
Ch 4-7 Uniform Circular Motion

• Uniform Circular Motion
• Motion with constant speed v in a circle of
  radius r but changing speed direction
• Change in direction of v causes radial or
  centripetal acceleration aR
• aR v2/r
• Period of motion T
• T (2?r)/v

16
Checkpoint 4-6

• Object has counterclockwise motion
• Then at y2 m , its velocity is v (4m/s)i
• and centripetal acceleration magnitude aRv2/R
  (4)2/2
• 8 m/s2
• aR(-8m/s2)j

An object moves at constant speed along a
circular path in a horizontal xy plane, with the
center at the origin. When the object is at
x-2m, its velocity is (4m/s)j. Give the object
(a) velocity and (b) acceleration when it is at
y 2m
17
Ch 4-8 Relative Motion in One Dimension

• Relative position
• xPAxPBxBA
• Relative Velocities
• d/dt(xPA)d/dt(xPB)d/dt (xBA)
• vPA vPB vBA
• Relative Acceleration
• d/dt(vPA)d/dt(vPB)d/dt (vBA)
• aPAaPB (vBA is constant)

18
Ch 4-8 Relative Motion in Two Dimensions

• Two observers watching particle P from origins of
  frames A and B, while B moves with constant
  velocity vbA with respect to A
• Position vector of particle P relative to origins
  of frame A and B are rPA and rPB. If rBA is
  position vector of the origin of B relative to
  the origin of A then
• rPA rPB. rBA and
• vPA vPB. vBA
• aPA aPB. Because vBA constant