Kinematic Physics for Simulation and Game Programming

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Kinematic Phsics for Simulation and Game Programming SI Phsics Units (International Sstem of Units) Quantit Units Linear position Meters Linear elocit Meters/second Linear acceleration Meters/second

0 International License Mike Baile mjb@cs.oregonstate.

Moment of Inertia Kilogram-meters Temperature º Celsius phsics-kinematic.

1 Kinematic Phsics for Simulation and Game Programming SI Phsics Units (International Sstem of Units) Quantit Units Linear position Meters Linear elocit Meters/second Linear acceleration Meters/second Force Newtons Energ Joules Power Watts Mass Kilograms Weight Newtons Densit Kilograms/meter This work is licensed under a Creatie Commons Attribution-NonCommercial- NoDeriaties 4.0 International License Mike Baile mjb@cs.oregonstate.edu Time Seconds Pressure Newtons/meter Momentum Kilograms-meters/second Angular position Radians Angular elocit Radians/second Angular acceleration Radians/second Moment (=torque) Newton-meters Moment of Inertia Kilogram-meters Temperature º Celsius phsics-kinematic.ppt mjb August 7, 08 mjb August 7, 08 SI Phsics Units (International Sstem of Units) Some Useful Conersions 4 Metrification of the World: meter = 9.7 inches =.8 feet mile =,60 meters =.60 kilometers mile per hour =.467 feet per second mile per hour = meters per second gallon =.79 liters cubic foot = 7.48 gallons = 8.5 liters kilogram =. pounds (mass, at Earth s surface) Newton = 0.4 pounds (force) pound = 4.45 Newtons (force) radian = 57.º US Metric Association mjb August 7, 08 mjb August 7, 08

What s the Difference Between Mass and Weight? 5 Some Useful Conersions 6 A gram is about the mass of a paper clip Mass is the resistance to acceleration and deceleration.

2 What s the Difference Between Mass and Weight? 5 Some Useful Conersions 6 A gram is about the mass of a paper clip Mass is the resistance to acceleration and deceleration. You can also think of it as inertia -- how difficult it is to accelerate a wagon with something in it. A nickel has a mass of about 5 grams Weight is the force pulling ou towards the center of whateer planetar bod ou happen to be standing on. A liter is half of a -liter soda bottle, or about a fourth of a gallon of milk A kilogram is a little more than twice as much as a pound (on Earth) On the moon, our mass would be the same as it is on Earth. It would still require the same amount of force to push ou in a (frictionless) wagon. A Newton is about ¼ of a pound On the moon, howeer, our weight would be about / 6 of what it is on Earth. Because most of us are stuck on Earth, within a mile or two of sea leel, in common practice, mass and weight designate about the same thing. A meter is a little more than a ard A kilometer is about 5 / 8 of a mile Water freezes at 0º Celsius A comfortable da is around 4º Celsius A reall hot da is around 5º Celsius mjb August 7, 08 Your bod temperature is about 7º Celsius mjb August 7, 08 Some Different Useful Conersions 7 Newton s Three Laws of Motion 8. Eer object in motion keeps that same motion (i.e., same speed and direction) unless an eternal force acts on it.. Force equals mass times acceleration (F=ma). For eer action, there is an equal and opposite reaction. F ma mjb August 7, 08 mjb August 7, 08

Acceleration Due to Grait Newton s Graitational Law sas that the attraction force between two objects is the product of their masses times the graitational constant G, diided b the square of the

3 Acceleration Due to Grait Newton s Graitational Law sas that the attraction force between two objects is the product of their masses times the graitational constant G, diided b the square of the distance between them: Earthradius F Gm m d where: G Nm kg and d is the distance between bod and bod For an object, m, at or near the surface of the Earth (i.e., d is the radius of the Earth) this simplifies to: meters feet F mg where: g 9.8. sec sec g is known as the Acceleration Due to (Earth s) Grait 9 0 at d d0 0t at Constant-Acceleration Formulas (these need to be memorized!) d d0 0t If ou are moing erticall, then the acceleration, a, will be the acceleration due to grait, g: +Y g = -9.8 meters/sec = -. feet/sec If ou are moing horizontall, there is no acceleration unless some outside horizontal force creates it. d d t 0 0 The following formula is hand because it relates all the usual quantities, but doesn t require ou to know the elapsed time: 0 a( dd0) -Y 0 mjb August 7, 08 mjb August 7, 08 cos,sin Projectile Motion A Projectile Launches Where Does it End Up? cos, sin d cliff h cliff Initial Quantities: X: Y: a 0 a g 0 cos 0 sin d cliff h cliff X: Quantities in Flight: d d t 0 0 Strateg: Treat each case separatel. Figure out what limitation makes the projectile stop moing, calculate the time to get to that, and then see where the projectile would hae ended up first. Y: h h0 0t mjb August 7, 08 mjb August 7, 08

4 Case. What if it Neer Reaches the Cliff? Case. What if it Hits the Side of the Cliff? 4 Y Distance Equation: t 0t 0 Note: g < 0. Sole for the time: t( 0) 0 * 0 t 0, g d cliff X Distance Equation: d cliff d 0. t cliff 0 Sole for the time: d * t t cliff 0 0 How Far will it Go? How High will it Go? Sole for the X Distance: d d t 0 0 * mjb August 7, 08 Sole for the Y Distance: h 0. 0t mjb August 7, 08 Case. What if it Lands on Top of the Cliff? 5 How do ou decide if Case,, or is the Correct Solution? 6 h impact Y Distance Equation: himpact 0. 0t 0t himpact 0 Sole for the time: Note: g < 0. d cliff h cliff At Bt C 0 t A B B 4AC Note: this will also work if h impact < 0. Note: if h impact == 0., this becomes Case # * 0 0 ghimpact t g How Far will it Go? Sole for the X Distance: d d t 0 0 * mjb August 7, 08 Consider Case #: If the horizontal distance that it traels is d cliff, then this is what happened. Consider Case #: If the height of the projectile is h cliff as its horizontal distance passes d cliff, then this is what happened. Otherwise, Case # is what happened. mjb August 7, 08 4

5 Case Projectile Motion: How Long will the Projectile Sta in Flight? 7 Case,, Projectile Motion: When and Where will the Projectile Reach its Maimum Height? 8 The elocit at the maimum height is zero d impact Wh are there solutions for t*? * 0 0 g( d0 dimpact ) t g h impact Y elocit Equation: 0 0 Sole for the time: t g g * 0 sin Note: g < 0. How do ou know which one is the correct one? How High will it Go? mjb August 7, 08 Sole for the Y Distance: * * * h 0. 0t g t mjb August 7, 08 Let s Tr it with Some Numbers 9. Now deal with the X component. What equation relates distance traeled to initial elocit and (zero) acceleration? 0 Find how much time it takes for the projectile to hit the ground. Find how far the projectile traels horizontall before hitting the ground. Find the maimum height the projectile reaches before starting back down. =(0.,0.) meters/sec Simplif g to be -0 meters/sec d d t Plug in the t ou got in step #. How far did the projectile trael? d 00t 5. Now deal with the maimum height. What is the Y elocit when the projectile reaches the maimum height?. Deal with just the Y component first. What is the equation that relates distance traelled to initial elocit and graitational acceleration? 0.0 dimpact d0 0t 6. What equation relates elocit achieed to initial elocit and distance traelled? (Hint: there is one that doesn t need t.) a( d d ) 0 0. Sole for t when d impact =d 0. Wh are there answers? What are the? Which one do ou want? d0 d0 0t 0t 7. Sole it for (d -d 0 ) ( d d ) 0 mjb August 7, 08 mjb August 7, 08 5

The Phsics of Bouncing Against a Floor or Wall A Ball Bouncing in a Bo To treat the case of an object bouncing against an immoeable object, such as a floor or a solid wall, the resulting elocit is: e

6 The Phsics of Bouncing Against a Floor or Wall A Ball Bouncing in a Bo To treat the case of an object bouncing against an immoeable object, such as a floor or a solid wall, the resulting elocit is: e Where e is the Coefficient of Restitution (CoR), and is a measure of how much energ is retained during the bounce. The amount of energ actuall retained during the collision is: Eretained e Which means that: Elost e Balls Bounced on a Concrete Surface: Ball Material CoR range golf ball tennis ball 0.7 billiard ball hand ball 0.75 wooden ball 0.60 steel ball bearing glass marble ball of rubber bands 0.88 hollow, hard plastic ball mjb August 7, 08 Current Position = (,) Current elocit = (, ) How long until the net bounce? t radius t floor radius left t radius right. Figure out which equation produces the minimum positie t alue. Adance the ball that much e. Perform the bounce in the proper direction 4. Handle the rest of the time step * g floor radius t g e or: If no bounce: t t mjb August 7, 08 The Phsics of Bouncing Against a Floor or Wall The Phsics of Bouncing Against a Floor or Wall 4 oid Bounce( float dt ) { while( dt > EPSILON ) { float tmin = dt; int which = NOTHING_HIT; float tleft =????; // time to hit the left wall if( tleft > 0. && tleft < tmin ) { tmin = tleft; which = HIT_LEFT; } // minimum time to do something // which reason was it for doing the something: t left radius left radius tleft float tright =????; // time to hit the right wall... float tfloor =????; // time to hit the floor... float tfloor =????; // time to hit the floor (note there are answers)... // tmin is now set to the smallest of:dt, tleft, tright, tfloor, tfloor which are all positie // which is set to: // NOTHING_HIT, HIT_LEFT, HIT_RIGHT, HIT_FLOOR, or HIT_FLOOR // to show what was the first thing hit mjb August 7, 08 // take a time step of time lenh tmin, using the projectile motion equations: // if a bounce is going to occur, tmin takes the ball right up to the surface: Xnow =????; t Ynow =????; t now =????; now =????; // change the proper elocit component: // if nothing was hit in this time step, just return: switch( which ) { case NOTHING_HIT: return; case HIT_LEFT:?????; break; case HIT_RIGHT: or:????? break; case HIT_FLOOR:????? break; case HIT_FLOOR:????? break; tmin dt } dt -= tmin; // after the bounce, we might still hae some time step left } } e e mjb August 7, 08 6

7 5 So far, we hae onl been dealing with objects undergoing linear motion What about objects undergoing rotational motion?. Spinning motion. Motion around a cure Spinning Motion: Constant-Acceleration Formulas 0 t 0 0t t ( ) 0 0 Angular acceleration (radians/sec ) Angular elocities (radians/sec) Note: In the same wa that the linear equations can work in,, and z, the rotational equations can work for rotations about the,, and z aes 6 mjb August 7, 08 mjb August 7, 08 Motion around a cure: Centripetal / Centrifugal Force 7 Then What is Centrifugal Force? 8. Eer object in motion keeps that same motion (i.e., same speed and direction) unless an eternal force acts on it.. Force equals mass times acceleration (F=ma). For eer action, there is an equal and opposite reaction. That force is called Centripetal Force, and acts on all objects as the round cures in order to make them round the cure and not go fling off For the car, that force is the friction of the road on the tires. For ou, that force is the side of the car on ou. mjb August 7, 08 The Centripetal Force is making the block change directions. This is what we obsere from the outside. But, if we were obsering while riding on the block, the motion direction that our mass wants to keep is the original straight line. But, the block has been forced to change to a new direction. Until our mass also changes to this new direction, we are going to keep tring to go in the old direction, which seems to us to be towards the outside of the cure. We think we are being thrown there, but it is just a natural consequence of Newton s First Law. mjb August 7, 08 7

Then What is Centrifugal Force? 9 How Much Force is the Centripetal Force and Where Does it Point? 0 r a r r s, ( meters, meters/sec ), s ( radians, radians/sec ) r r F ma m m r r http://kcd.

force. It is defined as a force, F, acting at a distance, d. (That distance is sometimes called a moment-arm.

8 Then What is Centrifugal Force? 9 How Much Force is the Centripetal Force and Where Does it Point? 0 r a r r s, ( meters, meters/sec ), s ( radians, radians/sec ) r r F ma m m r r mjb August 7, 08 This force points towards the instantaneous center of curature mjb August 7, 08 Torque Torque-Balance lets us Analze Mishaps Torque is like the rotational-motion equialent of force. It is defined as a force, F, acting at a distance, d. (That distance is sometimes called a moment-arm.) F A truck rounding a bend too quickl Center of Grait h Centrifugal Force = m R Piot Point d Grait Force (Weight) = mg Piot Point Torque = F d An object is in static equilibrium if :. Sum of the forces = 0. Sum of the torques = 0 The truck will tip if Tipping Torque > Grait Torque: w m h mg R mjb August 7, 08 What are the impacts of m,, R, h, and w? mjb August 7, 08 8

Kinematic Physics for Simulation and Game Programming

Kinematic Physics for Simulation and Game Programming Mike Bailey mjb@cs.oregonstate.edu physics-kinematic.pptx mjb October, 05 SI Physics Units (International System of Units) Quantity Units Linear position