Kinematic Physics for Simulation and Game Programming

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1 Kinematic Physics for Simulation and Game Programming Mike Bailey physics-kinematic.pptx mjb October, 05

2 SI Physics Units (International System of Units) Quantity Units Linear position Meters Linear velocity Meters/second Linear acceleration Meters/second Force Newtons Energy Joules Power Watts Mass Kilograms Weight Newtons Density Kilograms/meter 3 Time Seconds Pressure Newtons/meter Momentum Kilograms-meters/second Angular position Radians Angular velocity Radians/second Angular acceleration Radians/second Moment (=torque) Newton-meters Moment of Inertia Kilogram-meters 3 Temperature º Celsius mjb October, 05

3 SI Physics Units (International System of Units) Metrification of the World: US Metric Association mjb October, 05

4 Some Useful Conversions meter = inches = 3.8 feet mile =,60 meters =.60 kilometers mile per hour =.467 feet per second mile per hour = meters per second gallon = 3.79 liters cubic foot = 7.48 gallons = 8.35 liters kilogram =. pounds (mass, at Earth s surface) Newton = 0.4 pounds (force) pound = 4.45 Newtons (force) radian = 57.3º mjb October, 05

5 What s the Difference Between Mass and Weight? 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. Weight is the force pulling you towards the center of whatever planetary body you happen to be standing on. On the moon, your mass would be the same as it is on Earth. It would still require the same amount of force to push you in a (frictionless) wagon. On the moon, however, your 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 level, in common practice, mass and weight designate about the same thing. mjb October, 05

6 Some Useful Conversions A gram is about the mass of a paper clip A nickel has a mass of about 5 grams 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) A Newton is about ¼ of a pound A meter is a little more than a yard A kilometer is about 5 / 8 of a mile Water freezes at 0º Celsius A comfortable day is around 4º Celsius A really hot day is around 35º Celsius Your body temperature is about 37º Celsius mjb October, 05

7 Some Different Useful Conversions mjb October, 05

8 Newton s Three Laws of Motion. Every object in motion keeps that same motion (i.e., same speed and direction) unless an external force acts on it.. Force equals mass times acceleration (F=ma) 3. For every action, there is an equal and opposite reaction. F ma 3 mjb October, 05

Acceleration Due to Gravity Newton s Gravitational Law says that the attraction force between two objects is the product of their masses times the gravitational constant G, divided by the square of

9 Acceleration Due to Gravity Newton s Gravitational Law says that the attraction force between two objects is the product of their masses times the gravitational constant G, divided by the square of the distance between them: F Gm m d where: G Nm kg and d is the distance between body and body Earthradius For an object, m, at or near the surface of the Earth (i.e., d is the radius of the Earth) this simplifies to: F meters sec feet sec mg where: g g is known as the Acceleration Due to (Earth s) Gravity mjb October, 05

10 v v0 at d d0 v0t at Constant-Acceleration Formulas (these are worth memorizing!) dy dy0 vy0t gt If you are moving vertically, then the acceleration, a, will be the acceleration due to gravity, g: +Y g = -9.8 meters/sec = -3. feet/sec If you are moving horizontally, there is no acceleration unless some outside horizontal force creates it. d d v t x x0 x0 The following formula is handy because it relates all the usual quantities, but doesn t require you to know the elapsed time: v v0 a( dd0) -Y mjb October, 05

11 V Vcos,Vsin Projectile Motion d cliff h cliff Initial Quantities: a x X: Y: 0 vx0 V cos vy0 Vsin a y g X: Quantities in Flight: d d v t x x0 x0 Y: hy hy0 vy0t gt mjb October, 05

12 V A Projectile Launches Where Does it End Up? V cos,v sin d cliff h cliff Strategy: Treat each case separately. Figure out what limitation makes the projectile stop moving, calculate the time to get to that, and then see where the projectile would have ended up. mjb October, 05

13 Case. What if it Never Reaches the Cliff? V Y Distance Equation: v t gt y0 y0 gt v t 0 Solve for the time: t( gt v y 0) 0 t v * y 0 0, Note: g < 0. g How Far will it Go? Solve for the X Distance: d d v t x x0 x0 * mjb October, 05

14 Case. What if it Hits the Side of the Cliff? V X Distance Equation: d cliff 0. v t x0 d v t cliff x0 d cliff Solve for the time: t * d v cliff x0 How High will it Go? Solve for the Y Distance: hy 0. vy0t gt mjb October, 05

15 Case 3. What if it Lands on Top of the Cliff? V h impact 3 Y Distance Equation: himpact 0. vy0t gt gt vy0t himpact 0 Solve for the time: Note: g < 0. t At Bt C 0 4 A B B AC Note: this will also work if h impact < 0. t v * y0 vy0 ghimpact How Far will it Go? Solve for the X Distance: g Note: if h impact == 0., this becomes Case # d d v t x x0 x0 * mjb October, 05

16 How do you decide if Case,, or 3 is the Correct Solution? 3 h cliff d cliff Consider Case #: If the horizontal distance that it travels 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 #3 is what happened. mjb October, 05

17 Case 3 Projectile Motion: How Long will the Projectile Stay in Flight? V d impact t v * y0 vy0 g( dy0 dimpact ) g Why are there solutions for t*? How do you know which one is the correct one? mjb October, 05

18 Case,, 3 Projectile Motion: When and Where will the Projectile Reach its Maximum Height? V The y velocity at the maximum height is zero h impact Y Velocity Equation: v v gt y y0 0 Solve for the time: t v g V * y0 sin g Note: g < 0. How High will it Go? Solve for the Y Distance: h 0. v t g t * * * y0 mjb October, 05

19 Let s Try it with Some Numbers Find how much time it takes for the projectile to hit the ground. Find how far the projectile travels horizontally before hitting the ground. Find the maximum height the projectile reaches before starting back down. V=(0.,0.) meters/sec Simplify g to be -0 meters/sec. Deal with just the Y component first. What is the equation that relates distance travelled to initial velocity and gravitational acceleration? dimpact dy0 vy0t gt. Solve for t when d impact =d y0. Why are there answers? What are they? Which one do you want? dy0 dy0 0t 0t mjb October, 05

20 3. Now deal with the X component. What equation relates distance traveled to initial velocity and (zero) acceleration? d d v t x x0 x0 4. Plug in the t you got in step #. How far did the projectile travel? dx 00t 5. Now deal with the maximum height. What is the Y velocity when the projectile reaches the maximum height? What equation relates velocity achieved to initial velocity and distance travelled? (Hint: there is one that doesn t need t.) v v a( d d ) Solve it for (d -d 0 ) ( d d ) 0 mjb October, 05

21 The Physics of Bouncing Against a Floor or Wall To treat the case of an object bouncing against an immoveable object, such as a floor or a solid wall, the resulting velocity is: v ev Balls Bounced on a Concrete Surface: Ball Material CoR range golf ball tennis ball 0.7 Where e is the Coefficient of Restitution (CoR), and is a measure of how much energy is retained during the bounce. billiard ball hand ball 0.75 wooden ball The amount of energy actually retained during the collision is: Which means that: Eretained Elost e e steel ball bearing glass marble ball of rubber bands 0.88 hollow, hard plastic ball mjb October, 05

22 A Ball Bouncing in a Box Current Position = (x,y) Current Velocity = (v x,v y ) How long until the next bounce? x x v t x radius x left x x v t x radius x right y y vt y gt y floor radius t * v v g y y radius y y floor g. Figure out which equation produces the minimum positive t value. Advance the ball that much vx evx 3. Perform the bounce in the proper direction 4. Handle the rest of the time step vy vy vx vx v ev y or: y If no bounce: x x vxt y yvyt gt v v x y x v v gt y mjb October, 05

23 The Physics of Bouncing Against a Floor or Wall void Bounce( float dt ) { while( dt > EPSILON ) { float tmin = dt; int which = NOTHING_HIT; // minimum time to do something // which reason was it for doing the something: float tleft =????; // time to hit the left wall if( tleft > 0. && tleft < tmin ) { tmin = tleft; which = HIT_LEFT; } x v t x radius 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 positive // which is set to: // NOTHING_HIT, HIT_LEFT, HIT_RIGHT, HIT_FLOOR, or HIT_FLOOR // to show what was the first thing hit t left x left left x radius x v x mjb October, 05

24 The Physics of Bouncing Against a Floor or Wall // take a time step of time length tmin, using the projectile motion equations: // if a bounce is going to occur, tmin takes the ball right up to the surface: Xnow =????; Ynow =????; Vxnow =????; Vynow =????; // change the proper velocity component: // if nothing was hit in this time step, just return: x x vxt y yvyt gt vx v x v gt y y switch( which ) { case NOTHING_HIT: return; case HIT_LEFT:?????; break; case HIT_RIGHT:????? break; case HIT_FLOOR:????? break; case HIT_FLOOR:????? break; } tmin v ev x v v y v v x y y x v ev dt or: x y } } dt -= tmin; // after the bounce, we might still have some time step left mjb October, 05

25 So far, we have only been dealing with objects undergoing linear motion What about objects undergoing rotational motion?. Spinning motion. Motion around a curve mjb October, 05

26 Spinning Motion: Constant-Acceleration Formulas 0 t 0 0t t Angular acceleration (radians/sec ) Angular velocities (radians/sec) ( ) 0 0 Note: In the same way that the linear equations can work in x, y, and z, the rotational equations can work for rotations about the x, y, and z axes mjb October, 05

27 Motion around a curve: Centripetal / Centrifugal Force. Every object in motion keeps that same motion (i.e., same speed and direction) unless an external force acts on it.. Force equals mass times acceleration (F=ma) 3. For every action, there is an equal and opposite reaction. That force is called Centripetal Force, and acts on all objects as they round curves in order to make them round the curve and not go flying off For the car, that force is the friction of the road on the tires. For you, that force is the side of the car on you. mjb October, 05

28 Then What is Centrifugal Force? The Centripetal Force is making the block change directions. This is what we observe from the outside. But, if we were observing 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 trying to go in the old direction, which seems to us to be towards the outside of the curve. We think we are being thrown there, but it is just a natural consequence of Newton s First Law. mjb October, 05

29 Then What is Centrifugal Force? mjb October, 05

30 How Much Force is the Centripetal Force and Where Does it Point? a v r r s,v ( meters, meters/sec ), ( radians, radians/sec ) s r r v r v Fma m m r r This force points towards the instantaneous center of curvature mjb October, 05

31 Torque Torque is like the rotational-motion equivalent of force. It is defined as a force, F, acting at a distance, d. (That distance is sometimes called a moment-arm.) F Pivot Point d Torque = F x d An object is in static equilibrium if :. Sum of the forces = 0. Sum of the torques = 0 mjb October, 05

32 Torque-Balance lets us Analyze Common Mishaps A truck rounding a bend too quickly Center of Gravity w h Centrifugal Force = v m R Gravity Force (Weight) = mg Pivot Point The truck will tip if Tipping Torque > Gravity Torque: v w m h mg R What are the impacts of m, v, R, h, and w? mjb October, 05

Kinematic Physics for Simulation and Game Programming

Kinematic Physics for Simulation and Game Programming 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