Physics + Vectors

References:• xyz

Variable frame rates (review)

• Two options for handling it:– Option1: Cap frame rates• When moving / rotating express in units / updated• Good: Easy to understand• Bad: Slow machines can't reach the frame-rate

– Option2: Let computer run as fast as possible• When moving / rotating express in units / second

– The multiply by the time since the last frame

• Good: Runs on any speed machine• Bad: A little harder to understand

Velocity and Acceleration• Suppose you are moving at 50mph for 3

hours. How far are you from your original position?– A: 150 miles• 50 * 3

• Express this as a graph

– We're really calculating the area of this rectangle.

Speed (mph)

Time (hours)

50

3

Velocity and Acceleration, cont.

• What about (abruptly) changing velocity?

Speed (mph)

Time (hours)

50

3

Q: How far have we travelled after these 5 hours?Q (rephrased): In other words, what is the area of the pink area?A: 0.5*(50x1) + (50x3) + 0.5*(50x1) = 200 miles

Velocity and Acceleration, cont.

• we can approximate the area of this curve using (small) discreet time intervals

Speed (mph)

Time (hours)

50

3

A dT value for one iteration of the game loop.

import pygame# …clock = pygame.time.Clock()

#... (inside the game loop)dT = clock.tick() / 1000.0

Velocity and Acceleration, cont.

• What we're really doing is approximating the integral:

• Problem: This just tells us the distance we've moved, not the direction

• Q: What do we need to express distance and direction?

• A: Vectors!

Velocity and Acceleration

• represents a spaceship's position.• represents the ship's velocity (in units/s)• Suppose Δt seconds have passed• (the ship's new position) = ???

=

�⃗� 𝑣

𝑣∗ Δ𝑡

𝑝 ′

change in position due to velocity

Velocity and Acceleration, cont.• Velocity results in a

change in position• Acceleration results in

a change in velocity

Speed (mph)

Time (hours)

50

3

Position-offset (miles)

Time (hours)

25

3

50

75

100

Accel (m/h2)

3

50

𝒑 ′=�⃗�+∆ 𝒕 �⃗�

𝒗 ′=�⃗�+∆ 𝒕 �⃗�

Acceleration and Vectors, cont.• Our spaceship example, continued:– We fire the thrusters while facing in the direction – We hold the thrusters down for k (partial?)

seconds– The thrusters fire with a force of n units/s2

• Q: What is the new velocity of the ship (after these k seconds)

�̂��⃗�

𝑣 𝑛∗𝑘∗�̂�

𝑣 ′𝒗 ′=�⃗�+𝒏∗𝒌∗ �̂�

Gravity

• Gravity produces an accelaration on objects• Assuming we're on the earth, it points

downwards

𝑝1

�⃗�

𝑣1

�⃗�∆ 𝑡𝑣1 ′

𝑣1 ′ ∆ 𝑡

𝑝1 ′

Newton's Laws of Motion

1. The velocity of an object is constant unless acted upon by an outside force

2. For every action there is an equal-magnitude / opposite-direction action.

Simple collisions

• Not physically accurate• When we've looked at dot product, we'll re-

explore this and do proper– elastic collisions– inelastic collisions

• For now: – When two objects collide, impart a force• Equal magnitude, opposite direction on each body• Of a fixed magnitude

A new vector operator

• Not a standard one– Not the correct term, but I'll call it the tensor

product of two vectors.

– Not really any graphical interpretation.– Useful in:• Bouncing (next slide)• Lighting

Boundary collisions• We'll make this more flexible later. For now:– For each wall, generate a unit-length "normal", • Left is [1, 0] Right is [-1, 0]• Up is [0, 1] Down is [0, -1] (in pygame)

– When an object hits a wall add 2 * (.• Or to simulate friction, change the 2 to 1.9 (or similar)

�̂�

(.

2(

�⃗�

Gravity as a Force

• Gravity is a force– Proportional to the mass– But…when we apply the force• We divide by the mass (Newton's 2nd)• So…the acceleration is the same regardless of mass

– Unless there is air resistance (drag)

"Planetary" Gravity

• If we're simulating planets, gravity isn't "down"– Gravity is a force– Depends on the position of all objects • even if they're in another solar system!

– Where s is this object, i is the other object– In the real world, G = 6.6734^10-11 Nm/kg2

Examples / Labs

• #1:– Two bodies, exerting "planetary" gravity on the other.– Try to set up a semi-stable orbit.

• #2:– Create a bunch of circles, subject to gravity and bouncing

off walls• Bouncing off each other

– Click to lay a bomb.• #3:– Asteroids!– Bouncing asteroids– Realistic accelaration