Particle Kinematics: Translation at Constant Acceleration

1
Fac. of Comp., Eng. Tech. Staffordshire
University
Programming Physics Engines for Games
Particle KinematicsTranslation at Constant
Acceleration
Dr. Claude C. Chibelushi
2
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
• net external force
• implication of Newtons 2nd law of motion
• constant net force results in constant
  acceleration
• hence changing velocity

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Motion Formulas
Translation at constant acceleration
Displacement in equal time steps
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Motion Formulas
Summary of formula set
Constant net external force
Special case t1 0
Replace t by (t t1) in formulas above
General case
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Problem Solving Procedure

• Same steps as for constant velocity motion
• but different motion formulas
• 1D motion example
• Boy Racer Returns II
• Stafford to Stoke straight road, length 30 km
• car
• initial speed 60 km / h
• constant acceleration throughout race 5 km / h2
• Did boy racer break record (20 min)?

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Problem Solving Procedure

• 2D motion example
• Save Dino II
• What speed should Dino avoid?
• Initial motion parameters
• position (x, y) data
• dinosaur (10 km, 210 km). asteroid (10 km, 10
  km)
• asteroid speed 100 km / h. acceleration 2 km /
  h2
• Dinosaur speed constant

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Software Implementation

• Define data structures
• // C structure for particle in 2D space typedef
  struct particle2DTag
• Point2D pos
• Vector2D vel
• Vector2D acc
• Particle2D

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Software Implementation

• Declare variables
• particle position, velocity, acceleration
• Particle2D asteroid
• Point2D initPos
• simulation time
• long int relTime, initTime
• Initialise known quantities
• asteroid.pos.x 10.0 asteroid.pos.y 100.0
• asteroid.vel.x 710.7 asteroid.vel.y 10.7
• asteroid.acc.x 10.2 asteroid.acc.y 40.0

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Software Implementation

• Update simulation state (in simulation loop)
• alternative 1 direct application of position
  update formula
• relTime readComputerTime() - initTime
• asteroid.pos.x initPos.x asteroid.vel.x
  relTime asteroid.acc.x relTime relTime / 2
• asteroid.pos.y initPos.y asteroid.vel.y
  relTime asteroid.acc.y relTime relTime / 2

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Software Implementation

• Update simulation state (ctd.)
• alternative 2 for fixed time step
• multiplier for acceleration term is constant
  (hence better calculated outside simulation loop)
• float accMCoef timeStep timeStep / 2

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Software Implementation

• Update simulation state alternative 2 (ctd.)
• at each simulation clock tick
• update velocity
• asteroid.vel.x asteroid.acc.x timeStep
• asteroid.vel.y asteroid.acc.y timeStep
• update position
• asteroid.pos.x asteroid.vel.x timeStep
  asteroid.acc.x accMCoef
• asteroid.pos.y asteroid.vel.y timeStep
  asteroid.acc.y accMCoef

Only addition required if timeStep 1
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Software Implementation

• Food for thought
• Pros and cons of alternatives 1 and 2

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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

• Constant net external force, hence
• motion at constant acceleration
• equation of translational motion
• velocity update step proportional to time
  interval
• proportionality factor acceleration
• position update step has two components
• component proportional to time interval
  (proportionality factor initial velocity)
• component proportional to square of time interval
  (proportionality factor half of acceleration)

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Summary

• Problem solving procedure
• same steps as for constant velocity motion
• but different motion formulas
• Software implementation
• same as for constant velocity motion
• but data structures extension required to cater
  for acceleration
• repeatedly update simulation state (position and
  (possibly) velocity)