Motion of Objects in 2D and 3D

1
Motion of Objectsin 2D and 3D

• Topics in Computer Science
• Math Physics of Games
• Fall 2009

Some elements of this presentation are from
Stahlerr, some from Wikipedia, and some others
from MSDN
2
Some Vocabulary

• Speed is a scalar measuring rate of change in
  distance.
• Velocity is a vector measuring both speed and
  direction.
• Displacement is the distance traveled in a
  certain direction.
• Acceleration is the rate of change in velocity.
• Deceleration is a negative acceleration.

3
Displacement

• Distance
• D RxT
• Distance equals rate times time.
• Displacement
• D vt
• Displacement equals velocity times time.
• Which is measured in the following games
• Pole Position
• GTA
• Super Mario

4
Displacement in XNA

• Displacement with constant velocity
• Consider a game vehicle with constant velocity of
  150 px/s
• How far has the vehicle moved after 1 sec, 2
  sec, 3 sec, etc.
• D vt
• Displacements 150 px, 300 px, 450 px
• Some questions
• How do you represent this on screen?
• What moves? Vehicle or background?
• How far in each gameTime in XNA (60 cycles per
  sec) ?
• 150 px/sec / 60 sec 2.5 px
• What are the ramifications of 2.5 px per gameTime?

5
Average Velocity

• Calculate the difference in the displacement over
  time.
• Find the initial x-coordinate and subtract it
  from the final x-coordinate
• Divide by the time elapsed.
• V ?x xf xi t t
• Example
• Find average velocity in miles per hour
• ?x ¼ mile time 15 seconds
• V ¼ mi./ 15sec 0.25 /15 0.0167 mi/s
• Convert to mph
• 0.0167 mi/s 60 s/min 60 min/hr 60 mph

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Average Velocity Between Frames

• Suppose you set up the velocity of a UFO object
  to be
• ufo.position new Vector2(2.0f, 0.0f)
• What is the average velocity between two frames?
• ?x 2 px time 1/60 seconds
• V 2px/1/60 120 px per sec.
• Some questions
• Do you want your object to traverse 120 pixels
  each second?
• How long before it is off the screen?
• Would this simple calculation save you some time
  experimenting with velocities?
• If you were the designer, could you make this
  calculation without programming?

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Acceleration

• Acceleration rate of change of velocity
• May be constant, may vary.
• To calculate the acceleration, we are interested
  in the velocity as compared to time.
• a ? v vf vi ? t tf ti
• Example UFO travelling 1500 mph. After 5
  seconds, 3200 mph. What is the acceleration?
• A 3200 1500 340 mph. 5

8
Deceleration

• 2006 Bugatti Veyron 16.4 is clocked at 0 to 60 in
  2.65 s
• I was not so interested in a car that could go
  from 0 to 60 in 2.65 seconds, as one that
  could go from 60 to 30 in 0 seconds.
• Deceleration is negative acceleration.
• i.e. initial velocity is greater than the final
  velocity.

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Equations of Motion

• Final Velocity initial velocity acceleration
  time
• vf vi at
• Average velocity (initial velocity final
  velocity) / 2
• v (vi vf ) / 2
• Displacement ½ (initial velocity final
  velocity) time
• ?x ½ (vi vf )t
• Displacement initial velocity time ½
  acceleration time2
• ?x vit ½ at2
• Final velocity initial velocity2
  2accelerationdisplacement
• vf2 vi2 2a?x

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Gravity

• Acceleration due to gravity is 32 f/s2 or 9.8
  m/s2
• Drop a rock 50 m. How long till it hits the
  ground?
• vi 0
• vf we want
• a 9.8m/s2
• ?x -50m
• Which equation helps us here?

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Gravity

• Acceleration due to gravity is 32 f/s2 or 9.8
  m/s2
• Drop a rock 50 m. How long till it hits the
  ground?
• vi 0
• vf we want
• a 9.8m/s2
• ?x 50m
• Which equation helps us here?
• 4 has all the elements in it. We can rearrange
  it to find time.
• ?x vit ½ at2
• 50 m 0t ½(9.8m/s2)t2
• 50 4.9t2
• t2 50/4.9 10.20s2
• t 3.19s

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BREAK! GASP!
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Motion in 1, 2 and 3 Dimensions

• 1D displacement along a line
• 2D displacement across a plane
• 3D displacement through space

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Displacement in 2D

• ?r rf ri for position vectors ri (initial)
  and rf (final).
• Object moves from point P (50, 400) to P
  (550, 100)
• Express as vectors
• P 50 400
• P 550 100
• Solve geometrically
• Draw both vectors from the origin. Draw P P
• Solve algebraically
• Subtract the two vectors P P to find resultant
  vector ?r
• P P 550 100 - 50 400 500 -300
• Displacement is 500 pixels left and 300 pixels
  down
• Similarly in 3D

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Average Velocity in 3D

• Object moves from P(150, 0, 250) to P(400, 250,
  -300) in 5 seconds.
• Calculate displacement
• PP 150 0 250 400 250 -300 -250 -250
  550
• Calculate average velocity
• V ?r -250 -250 550 50 50 -110 t
  5

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Equation of Motion for 2D 3D

• For vectors a, vi, vf, and ?r and scalar t
• Final Velocity initial velocity acceleration
  time
• vf vi at
• Displacement ½ (initial velocity final
  velocity) time
• ?r ½ (vi vf )t
• Displacement initial velocity time ½
  acceleration time2
• ?r vit ½ at2

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Projectiles

• Objects set in motion affected by gravity move in
  parabolic path due to the acceleration due to
  gravity.
• a vt ½ gt2
• NOTE Horizontal motion is independent of
  Vertical motion.
• So we calculate the coordinates separately
• Objects set in motion at a given angle
• Initial velocity (d cos ? , d sin ? )
• Acceleration and air friction due to gravity
  affects vertical vectors
• Air friction affects horizontal vectors

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Independent Study

• Work through examples of projectile motion in
  Chapter 10, pp.296-310
• Some questions to ask yourself repeatedly
• Where did equation come from?
• Are we looking at horizontal or vertical motion?