Physics in Game
Programming
Basic Physics Review
Vector: a quantity that has two independent properties: magnitude
and direction. Examples of vectors in nature are velocity,
acceleration, momentum, force, electromagnetic fields, and weight.
Velocity: change in position over time.
speed: mph, fps, pixels per frame
angle: degrees, radians
speed
angle (dir)
Basic Physics Review
Vector: a quantity that has two independent properties: magnitude
and direction. Examples of vectors in nature are velocity,
acceleration, momentum, force, electromagnetic fields, and weight.
Velocity: change in position over time.
(6,8)
y
speed
angle (dir)
(0,0)
x
(x, y)
Vector = (6, 8)
Using trig the quantities can be converted
from one format to another.
Velocity Example
Velocity: change in position over time.
vel(x, y) = (5, 0)
Unit of measurement?
• MPH
• FPS
• Pixels per frame
• Pixels per second
vx = 5
vy = 0
(0,0)
(5,0)
(10,0)
(15,0)
(20,0)
(25,0)
(30,0)
(35,0)
(40,0)
Add velocity values to current location (x, y) every cycle.
Acceleration
Acceleration: change in velocity over time.
angle: degrees, radians
Velocity
angle (dir)
Acceleration
Vector: a quantity that has two independent properties: magnitude
and direction..
Acceleration: change in velocity over time.
(6,8)
velocity
y
angle (dir)
(0,0)
x
(x, y)
Vector = (6, 8)
Using trig the quantities can be converted
from one format to another.
Velocity without Acceleration
initial velocity
new velocity
vel(x,y) = (0, 0)
vel(x,y) = (50, 0)
(0, 0)
(50, 0)
(0, 0)
(50, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – mph2
(5, 0)
vel(x,y) - mph
(0, 0)
position(x,y) - feet
(0, 0)
0 seconds
(0, 0)
(58, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – mph2
(5, 0)
(5, 0)
vel(x,y) - mph
(0, 0)
(5, 0)
Displacement formula
s = ut + 1/2at2
s = distance, u = initial velocity
position(x,y) - feet
(0, 0)
(4, 0)
Miles to Ft Conversion
ft = miles * 5280ft
1 second
(4, 0)
.00069 miles
(58, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – mph2
(5, 0)
(5, 0)
(5, 0)
vel(x,y) - mph
(0, 0)
(5, 0)
(10, 0)
position(x,y) - feet
(0, 0)
(4, 0)
(15, 0)
2 seconds
(15, 0)
.00278 miles
(58, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – mph2
(5, 0)
(5, 0)
(5, 0)
(5, 0)
vel(x,y) - mph
(0, 0)
(5, 0)
(10, 0)
(15, 0)
position(x,y) - feet
(0, 0)
(4, 0)
(15, 0)
(33, 0)
3 seconds
(33, 0)
.00625 miles
(58, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – mph2
(5, 0)
(5, 0)
(5, 0)
(5, 0)
(5, 0)
vel(x,y) - mph
(0, 0)
(5, 0)
(10, 0)
(15, 0)
(20, 0)
position(x,y) - feet
(0, 0)
(4, 0)
(15, 0)
(33, 0)
(58, 0)
4 seconds
(58, 0)
.01111 miles
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – pps2
(5, 0)
vel(x,y) - pps
(0, 0)
pps = pixels per second
position(x,y) - pixels
(0, 0)
0 seconds
(0, 0)
(50, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – pps2
(5, 0)
(5, 0)
Add
vel(x,y) - pps
(0, 0)
(5, 0)
pps = pixels per second
Add
position(x,y) - pixels
(0, 0)
(5, 0)
1 second
(5, 0)
(50, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – pps2
(5, 0)
(5, 0)
(5, 0)
Add
vel(x,y) - pps
(0, 0)
(5, 0)
(10, 0)
pps = pixels per second
Add
position(x,y) - pixels
(0, 0)
(5, 0)
(15, 0)
2 seconds
(15, 0)
(50, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – pps2
(5, 0)
(5, 0)
(5, 0)
(5, 0)
Add
vel(x,y) - pps
(0, 0)
(5, 0)
(10, 0)
(15, 0)
pps = pixels per second
Add
position(x,y) - pixels
(0, 0)
(5, 0)
(15, 0)
(30, 0)
3 seconds
(30, 0)
(50, 0)
Acceleration
Acceleration: change in velocity over time.
accel(x,y) – pps2
(5, 0)
(5, 0)
(5, 0)
(5, 0)
(5, 0)
Add
vel(x,y) - pps
(0, 0)
(5, 0)
(10, 0)
(15, 0)
(20, 0)
pps = pixels per second
Add
position(x,y) - pixels
(0, 0)
(5, 0)
(15, 0)
(30, 0)
(50, 0)
4 seconds
(50, 0)
Comparison
Feet Per Second
(0, 0)
(4, 0)
(15, 0)
(33, 0)
(58, 0)
4 seconds
Pixels Per Second
(0, 0)
(5, 0)
(15, 0)
(30, 0)
(50, 0)
4 seconds
Conversion Formula1
I began this lesson by defining a vector as a quantity that has two
independent properties: magnitude and direction. I pointed out that
with some basic trigonometry these two properties could be
converted into x and y coordinate properties. Below is the formula for
converting the properties magnitude and direction into x and y
properties.
dx = magnitude • cos(direction)
dy = magnitude • sin(direction)
dx = magnitude * Math.cos(Math.toRadians(direction));
dy = magnitude * Math.sin(Math.toRadians(direction));
Conversion Formula2
Here is the formula for converting a vector represented by x
and y properties into magnitude and direction properties.
𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = atan 𝑑𝑦, 𝑑𝑥
𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 = 𝑑𝑥 ∙ 𝑑𝑥 + 𝑑𝑦 ∙ 𝑑𝑦
Pythagorean Theorem
direction = (int)Math.toDegrees(Math.atan2(dy, dx));
magnitude = Math.sqrt(dx*dx + dy*dy);
Vectors
We Don’t Need Know Stinking Vectors
Since a vector is simply a way to represent two quantities, it
turns out that we don’t even need to use vectors in our game
programming. Instead of representing velocity as a vector we
can just create two variables named vx and vy to represent
velocity. Instead of representing acceleration as a vector we can
create two variables named ax and ay to represent acceleration.
private
private
private
private
int
int
int
int
vx;
vy;
ax;
ay;
//
//
//
//
horizontal velocity
vertical velocity
horizontal acceleration
vertical acceleration
Summary
Velocity and Acceleration are two very important properties in
game programming because they allow you to animate objects
in a way that mimics real life. Fortunately we can represent these
properties in games without having to use a lot of advanced
math but still having a precision that is close enough to give the
illusion that these properties behave just like their real world
counterparts.
private
private
private
private
int
int
int
int
vx;
vy;
ax;
ay;
//
//
//
//
horizontal velocity
vertical velocity
horizontal acceleration
vertical acceleration
Summary
To add velocity and acceleration to an object in a game simply add the
acceleration values to the velocity values and then add the velocity
values to the object’s (x, y) position during each time cycle of the game.
private int x = 0;
// horizontal position
private int y = 0;
// vertical position
private int vx = 5;
// horizontal velocity
private int vy = 5;
// vertical velocity
private int ax = 2;
// horizontal acceleration
private int ay = 2;
// vertical acceleration
-----------------------------------------------------------vx = vx + ax; vy = vy + ay;
// add accel to vel
x = x + vx;
y = y + vy;
// add vel to current position