# Phun with Physics The basic ideas Vector calculus

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Phun with Physics The basic ideas Vector calculus
Mass, acceleration, position, …

Some defs… Kinematics Dynamics The status of an object
Position, orientation, acceleration, speed Describes the motion of objects without considering factors that cause or affect the motion Dynamics The effects of forces on the motion of objects.

The basics Let p(t) be the position in time. Other values:
We’ll drop the (t) and just say p Other values: v – velocity a - acceleration Velocity is the derivative of position Acceleration is the derivative of velocity

Vector calculus Really, p(t) is a triple, right?
Think of these equations as three equations, one for each dimension This is vector calculus

Newton’s law and Momentum
We know this one… Momentum Note: Force is the derivative of momentum

Think of a real object as a bunch of points, each with a momentum We can find a “center of mass” for the object and treat the object as a point with the total mass We have: Mass, position vector, acceleration vector, velocity vector

Example: Air Resistance
The resistance of air is proportional to the velocity F = -kv We know F = ma, so: ma = -kv So, how can we solve?

Euler’s method

We’ll start in 2D… Let W be the orientation (angle) Easy in 2D, not so easy in 3D we’ll be back w is the angular velocity around center of gravity a is the angular acceleration Then…

What’s the velocity at a point?
Radians are important, here Perpendicular to cp Same length! Perpendicularized radius vector p Chasles’ Theorem: Velocity at any point is the sum of linear and rotational components.

Angular momentum Angular momentum At a point? For the whole thing?
M=mv Momentum equals mass times velocity What does this mean?

We want to know how much of the momentum is “around” the center
Angular Momentum of a point around c rcp

Torque – Angular force Torque at a point

Total angular momentum and moment of inertia
I is the moment of inertia for the object.

Moment of inertia What are the characteristics?
What does large vs. small mean? How to we get this value?

Relation of torque and moment of inertia

Simple dynamics Calculate/define center of mass (CM) and moment of inertia (I) Set initial position, orientation, linear, and angular velocities Determine all forces on the body Linear acceleration is sum of forces divided by mass Angular acceleration is sum of torques divided by I Numerically integrate to update position, orientation, and velocities

Object collisions Imagine objects A and B colliding
Assume collision is point on plane A n

What happens? ??? B A n

Our pal Newton… Newton’s law of restitution for instantaneous collisions with no friction e is a coefficient of restitution 0 is total plastic condition, all energy absorbed 1 is total elastic condition, all energy reflected What’s the consequence of “no friction”?

Impulse felt by each object
Newton’s third law: equal and opposite forces Force on A is jn (n is normal, j is amount of force) Force on B is –jn So… B We’ll need to know j A n No rotation for now…

Solving for j… Then plug j into…

How to determine the time of collision?
What do I mean? Ideas? Avoid “tunneling”: objects that move through each other in a time step

Euler’s method We can write…

The problem with Euler’s method
Taylor series: We’re throwing this away! Error is O(Dt2) Half step size cuts error to ¼!

Improving accuracy Supposed we know the second derivative?
But, we would like to achieve this error without computing the second derivative. We can do this using the Midpoint Method

Midpoint Method 1. Compute an Euler step 2. Evaluate f at the midpoint
3. Take a step using the midpoint value.

Example Step 1 Step 2 Step 3

Moving to 3D Some things remain the same The killer: Orientation
Position, velocity, acceleration Just make them 3D instead of 2D The killer: Orientation “It’s possible to prove that no three-scaler parameterization of 3D orientation exists that doesn’t ‘suck’, forsome suitably mathematically rigorous definition of ‘suck’.” Chris Hecker

Orientation options Quaternions Matricies We’ll use later
We’ll use this here.

Variables x(t) v(t) q(t) Spatial position in time (3D)
Center of mass at time t Assume object has center of mass at (0,0,0) v(t) Velocity in space q(t) Orientation in time (quaternion)

Angular velocity w(t) Angular velocity at any point in time
This is a rotation rate times a rotation vector Strange, huh?

Note Angular velocity is instantaneous
We’ll compute it later in equations, but we won’t keep it around. Angular velocity is not necessarily constant for a spinning object! Interesting! Can you visualize why?

Angular velocity and vectors
Rate of change of a vector is: This is why they like that notation Change is orthogonal to the normal and vector Magnitude of change is vector length times magnitude of w

Derivative of the rotation quaternion
Woa… Normalize this sucker and we can take a Newton step. Be sure to scale w(t) by Dt.

Velocity of a point (think vertex)
Not just v(t), must include rotation!!! Position of point at time t: Just a reminder: Velocity of point at time t: And the magic: Angular part Linear part

Force and torque Fi(t) Force on particle i at time t Vector, of course
Torque on point i: Total Force: Total Torque:

Linear Momentum Momentum: And, the derivative of momentum is force:

Angular Momentum Angular momentum is preserved if no torque is applied. L(t) is the angular momentum

Inertia Tensor All that work typing this sucker in and it’s pretty much useless. Orientation dependent!

Inertia Tensor for a base orientation

Base inertia tensor to current inertia tensor
Relation of angular momentum and inertia tensor:

Bringing it all together
Current state: Position Orientation Linear momentum Angular momentum Velocity (may itself be updated)

3D Collisions There are now two possible types of collisions
Not just vertex to face

3D collisions Vertex to face Edge to edge

We need… 3D collision detection Determine when the collision occurred
We’ll do that later Determine when the collision occurred Binary search for the time Bisection technique

Velocity at a point Position (center of gravity) Position of point
Angular velocity Velocity (linear)

Relative velocity Normal to object b Vertex to face: normal for face
Edge to edge: cross product of edge directions Positive vrel means moving apart, ignore it Negative vrel means interpenetrating, process it What does zero vrel mean?

New velocities for a and b
Vector from center of mass to point Let: Normal Velocity (linear) update Mass Angular velocity update Inertia tensor inverse Equivalent equations for b

And j…

Contrast 2D/3D 2D 3D

Questions? 2D had two different j’s 3D gets by with only one.
Linear velocity and angular velocity 3D gets by with only one. Why is this?

Other issues Resting contact
Bodies in contact, but –e < vrel < e Must deal with motion transfer