Basic Motion

Object motion is represented with vectors.
Velocity is a vector:
   Vector direction is direction of movement
   Vector magnitude is speed of movement

Velocity vector corresponds to amount object will move in one unit of time.

Basic Motion

Displacement = Velocity * time

If an object starts at position P0, with velocity V,
after t time units, its position P(t) is:

            P(t) = P0 + V * t

Note: choice of units is arbitrary, as long as things are consistent.
e.g. use meters for distance, seconds for time, and meters/second for velocity.

Don't try to combine meters/second with miles/hour, for instance.

Varying Velocity

The previous formula only works if the object moves with a constant velocity.

In many cases, objects' velocities change over time.

In such a case, velocity is a function that we integrate.

Displacement = Velocity dt

In complex motion, there isn't an analytical solution (i.e. a simple formula).

Euler Integration

Euler integration approximates an integral by step-wise addition.

At each time step, we move the object in a straight line using the current velocity:

    dt = t1 - t0
    P(t1) = P(t0) + V * dt



Velocity is the integral of acceleration:

Velocity = Acceleration dt

Applying Euler integration again gives:

    dt = t1 - t0
    Acc = computeAcceleration()
    Vel = Vel + Acc * dt
    Pos = Pos + Vel * dt


Gravity near the Earth's surface produces a constant acceleration of 9.8 m/sec2

In this case:

Acc = Vector([0, -9.8, 0])



Newton's 2nd Law of Motion:

Force = Mass * Acceleration

Which can be rewritten as

Acceleration = Force / Mass

If an object has mass M, and force F is applied to it,
its motion can be calculated via Euler integration:

    Acc = F / M
    Vel += Acc * dt
    Pos += Vel * dt

Note that F, Acc, Vel, and Pos are all vectors. M is a scalar.

Gravity Again

F = (G * M1 * M2) / d2

For a complete simulation, we need to calculate the force on each object every frame.

When multiple forces are applied, their vectors are added.


Drag (slow moving objects):

F = Cdrag * V

Drag (fast moving objects):

F = Cdrag * V2


F = ρliquid * g * Volume