Motion in Two and three dimensions

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Motion in Two and three dimensions

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Position and Displacement

• The position vector of an object with coordinates
  (x,y,z) can be written as rxiyjzk
• i,j,k are unit vectors dimensionless vectors
  with a magnitude one that point in the
  direction of the x, y, and z axis respectively
• Example
• position vector r5i10j0k corresponds with
  the coordinates (5,10)

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Position and Displacement continued

• Displacement
• ?r r2-r1
• In unit vector notation
• ?r(x2x1)i(y2y1)j(z2z1)k
• or ?rxiyjzk

?
?
?
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This slide needs to be fixed!!
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Average and Instantaneous Velocity

• Average velocity ?r/?tdisplacement
• 
  Interval of time
• In vector components
• Vavg. (?x/?t)i (?y/?t)j(?z/ ?t)k
• Instantaneous velocity-as time interval shrinks
  to zero
• Vdr/dt

?
?
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Velocity continued

• The direction of the instantaneous velocity of a
  particle is always tangent to the particles path
  at its position.
• In unit vector notation
• v(dx/dt)i(dy/dt)j(dz/dt)k

?
This slide needs an illustration!
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Average Acceleration and Instantaneous
Acceleration

• Average accelerationchange in velocity
• change in
  time
• As ?t approaches 0 you get instantaneous
  acceleration adv/dtd2r/dt2
• In unit vector notation
• aaxiayjazk

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Projectile Motion

• Projectile- a particle that moves in a vertical
  plane with some initial velocity but its
  acceleration is always the free-fall acceleration
  (downward)
• Initial velocityvoxivoyj
• Components voxand voy can be found if the angle
  between the initial velocity and the positive
  x-axis
• voxvocos? and voy vosin?

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Projectile Motion Continued

• In projectile motion the horizontal and vertical
  motion are independent of each other.
• Projectile motion problems can be solved by
  dividing the problem into horizontal motion
  (accelerationzero, velocity is constant) and
  vertical motion (constant downward acceleration)

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Projectile Motion Continued

• Horizontal Motion
• Vx remains unchanged from vi
• dhx-xivixt (vixvicos?)
• dh(vicos?)
• Vertical Motion
• dv y-yiviyt-(1/2)gt2(visin?)t-(1/2)gt2

Can you find the small mistake on this slide?
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Uniform Circular Motion

• A particle is in uniform circular motion if it
  travels around a circle or a circular arc at
  uniform speed
• Although there is no change in speed the particle
  is constantly accelerating because there is
  constantly change in direction
• Centripetal motion-acceleration is always
  directed radially inward
• av2/r T2pr/v

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Relative Motion in One and Two Dimensions

• The velocity of a particle depends on the
  reference frame
• Usually velocity is measured relative to the
  ground
• xbcxbaxac (two outside subscripts)
• rbcrbarac
• vbcvbavac

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Comprehension Questions

• What is the horizontal velocity at the top of the
  trajectory?

Vx
Make this problem more specific!
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• What is the vertical component of velocity at the
  top of the trajectory?

0 m/s
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Comprehension Questions

• In projectile motion are the horizontal and
  vertical components dependant on each other? Why
  or why not?

No, because there is an acceleration in only one
of the components and gravity affects only one of
the components.
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• Gravity affects which velocity component of
  projectile motion?

vertical component
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• Which velocity component has constant
  acceleration and in what direction is the
  magnitude?

vertical component due to gravity in the downward
direction
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• What is the horizontal acceleration component of
  an object in projectile motion?

0 m/s2
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• Does an object in uniform circular motion have an
  acceleration?

yes
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• Is that acceleration caused by a change in speed?
  If not what is it caused by?

no, there is a change in direction causing a
change in velocity which means there is an
acceleration
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• What is the formula for centripetal acceleration?

a v2/a
Fix the mistake!
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• What is the formula for centripetal force?

Fc mv2/r
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• What is the formula for the period of revolution?

T2pr/v
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• In what direction does the centripetal
  acceleration point?

towards the center
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• What direction is the centripetal force pointing
  towards?

towards the center
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• In what direction relative to the centripetal
  acceleration does the velocity always point?

perpendicular to the accceleration
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• In what direction relative to the circle does the
  velocity always point?

tangent to the circle
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• What are the vector components of the velocity?

v vxi vyj
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• What are the vector components of centripetal
  acceleration?

a axi ayj
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• What is the most common reference frame?

ground
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• What is the equation for relative motion?

XP-A XP-B XB-A
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• What is the equation for average velocity and
  instantaneous velocity?

vavg change in r / change in t v dr/dt
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• What are the formulas for average acceleration
  and instantaneous acceleration?

aavg change in v / change in t a dv/dt
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Sample questions

• The original position of a particle is
  r(-3m)i (2m)j(5m)k, and then the position is
  r2(9m)i(2m)j(8m)k?

What is the question??
34

• A particle with velocity v0-2i4j ( in meters
  per second) at t0 undergoes a constant
  acceleration of magnitude 30m/s2 at an angle of
  130 degrees from the positive x-axis. What is the
  particles velocity at t5 s in unit vector
  notation?

Good questionalso find the displacement between
3 and 5 seconds.
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• A particle moves so that its position ( in
  meters) as a function of time is r i4t2jtk.
  Write expressions for its acceleration and
  velocity as functions of time.

Got an answer for this one?
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• A car moving at a constant 60km/h moves east for
  40 min. then in a direction 50 degrees east of
  due north for 20 min. and then west for 50min.
  What are the magnitude and angle of its average
  velocity?

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• A small ball rolls horizontally off the edge of a
  table that is 1.20 m high. It hits the floor at
  1.52 m from the edge. How long was it in the air?

Also find the speed and angle at which it hits!!
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• A rotating fan completes 1200 rev. per minute.
  The radius of the blade is 0.15m. What distance
  does the tip move in one rev.?

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• A fly ball has v 25 i 4.9j ( in meters per
  second). Has the ball passed its highest point?

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• A particle has a displacement of ?r 2.0 i-3.0 j
  6.0 k and has a final position of r3.0j-4.0k.
  What was its initial position vector?

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• A ball has an initial speed of 25m/s.what are the
  least and greatest elevation angles at which the
  ball can be kicked to score a field goal from a
  point 50 min front of the goal post whose
  horizontal bar is 3.44 m above the ground.

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• A projectiles launch speed is 5X the speed at
  its max height. Find the launch angle.

Ok, these are good questions theohow about
solutions so people can check their answers?