Particle Kinematics: Translation at Constant Velocity

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Fac. of Comp., Eng. Tech. Staffordshire
University
Programming Physics Engines for Games
Particle KinematicsTranslation at Constant
Velocity
Dr. Claude C. Chibelushi
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Outline

• Introduction
• Motion Formulas
• Problem Solving Procedure
• Software Implementation
• Summary

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Introduction

• This lecture considers rigid body which has
• no (or negligible) angular motion
• only translational motion considered
• particle model can be used
• no net external force

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Introduction

• Implication of Newtons 1st and 2nd law of
  motion
• no net force results in no acceleration
• hence constant velocity
• constant magnitude (may be 0)
• constant direction
• hence rectilinear motion

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Motion Formulas
Translation at constant velocity
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Motion Formulas

• Average speed
• distance travelled divided by corresponding
  duration
• For rectilinear motion distance travelled equals
  displacement magnitude
• hence, during time interval ?t t2 t1
• for displacement ?s s2 s1, average velocity
  v ?s / ?t
• If velocity is constant
• average velocity equals instantaneous velocity

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Motion Formulas
Rearranging gives
Generalising p2 to p, and t2 to t (i.e. p is
position at any instant t) gives
If t1 0 (e.g. stop watch reset to 0 at instant
t1) then
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Motion Formulas
Summary of formula set
No net external force
General case
Special case t1 0
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Problem Solving Procedure

• 1D motion
• Specify coordinate system
• origin
• positive and negative direction along axis of
  motion
• e.g. corresponding to forward or backward movement

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Problem Solving Procedure

• 1D motion
• Specify known motion parameters
• use signed scalar values
• Compute unknown motion parameter(s) using
  appropriate formula(s)

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Problem Solving Procedure

• 1D motion example
• Develop physics engine for coolest game ever
• Boy Racer Returns!

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Problem Solving Procedure

• 1D motion example
• Boy racing car along straight road from Stafford
  to Stoke
• road length 30 km
• car speed at departure point 60 km / h
• no net external force acts on car and driver
  throughout race
• challenge break the record (20 min)
• Did boy racer win?

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Problem Solving Procedure

• 2D motion
• Specify coordinate system
• origin
• positive and negative direction of x- and
  y-coordinate axes, corresponding to e.g.
• forward or backward movement
• up or down movement

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Problem Solving Procedure

• 2D motion
• Specify known motion parameters
• two signed scalar values for each 2D parameter
• Compute unknown motion parameter(s) using
  appropriate formula(s)
• can apply formula(s) to each axis, independently
  of other axis

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Problem Solving Procedure

• 2D motion example
• Develop physics engine for VR application Save
  Dino
• the last surviving dinosaur is running for its
  life, in a hail of asteroids
• Dinos path crosses asteroid path
• player sets asteroid velocity
• computer advises Dino what speed to avoid

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Problem Solving Procedure

• 2D motion example
• Initial motion parameters
• position (x, y) data
• dinosaur (10 km, 210 km)
• asteroid (10 km, 10 km)
• asteroid speed 100 km / h (constant)
• Dinosaur speed constant
• challenge save Dino! Tell Dino what speed to
  avoid

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Software Implementation

• Define data structures
• possible data grouping
• // C structure for 2D point or vector
• typedef struct vector2DTag
• float x
• float y
• Point2D, Vector2D
• note structure replaced by class in Java

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Software Implementation

• possible data grouping (ctd.)
• // C structure for particle in 2D space
• typedef struct particle2DTag
• Point2D pos
• Vector2D vel
• Particle2D

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Software Implementation

• Declare variables
• particle position, velocity
• Particle2D asteroid
• simulation time
• long int curTime

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Software Implementation

• Initialise known quantities
• asteroid.pos.x 10000.0
• asteroid.pos.y 10000.0
• asteroid.vel.x 70710.7
• asteroid.vel.y 70710.7

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Software Implementation

• Update simulation state (in simulation loop)
• curTime readComputerTime()
• timeStep curTime - prevUpdateTime
• prevUpdateTime curTime
• asteroid.pos.x asteroid.vel.x timeStep
• asteroid.vel.y asteroid.vel.y timeStep

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Software Implementation

• Update simulation state (ctd.)
• simplification for fixed time step
• position increment / decrement at each
  simulation clock tick is constant (calculated
  once outside simulation loop)
• Vector2D posStep
• posStep.x asteroid.vel.x timeStep
• posStep.y asteroid.vel.y timeStep
• update position (inside simulation loop)
• asteroid.pos.x posStep.x
• asteroid.vel.y posStep.y

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Suggested Reading

• Relevant parts of Ch. 2, D.M. Bourg, Physics for
  Game Developers, OReilly Associates, 2002.

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Summary

• No net external force, hence
• no motion
• or rectilinear motion at constant velocity
• equation of translational motion position update
  step proportional to time interval
• proportionality factor velocity

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Summary

• Problem solving procedure
• specify coordinate system, and known motion
  parameters
• compute unknown motion parameter(s)
• Software implementation
• define data structures
• possibly group related variables
• declare variables and initialise known quantities
• repeatedly update simulation state