Maths behind Unity rigidbody.AddForce

Question

I've tried to understand the Maths behind AddForce, as to reproduce it using rigidbody.velocity +=.

Based on the formulas provided by PhysX documentation for AddForce, we have:

eFORCE            parameter has unit of mass * distance/ time^2, i.e. a force
eIMPULSE          parameter has unit of mass * distance /time
eVELOCITY_CHANGE  parameter has unit of distance / time, i.e. the effect is mass independent: a velocity change.
eACCELERATION     parameter has unit of distance/ time^2, i.e. an acceleration. It gets treated just like a force except the mass is not divided out before integration.

I tried to convert these formulas above to a rigidbody.velocity += syntax, but I am unable to get the Maths right.

Here is what I got working:

float force = 3f;
public void FixedUpdate() {
  // Same as r.AddForce(force, ForceMode.Force)
  Vector2 distance = (force / r.mass) * Time.fixedDeltaTime;
  r.velocity += distance;
} 

float force = 3f;
public void FixedUpdate() {
  // Same as r.AddForce(force, ForceMode.Impulse)
  Vector2 distance = (force / r.mass);
  r.velocity += distance;
} 

I tested both of them, by superposing two rigidbodies, one using AddForce, while the other used r.velocity+=. So far, it seem my convertion is correct as they move exactly the same.

As for the Maths I am not getting right, here is what I came with:

eFORCE = mass * distance/ time^2
eFORCE / mass = distance/ time^2
(eFORCE / mass) * time^2 = distance

distance = (eFORCE / mass) * time^2

Same as ?

Vector2 distance = (force / r.mass) * Time.fixedDeltaTime;

That woud mean time^2 is equal to Time.fixedDeltaTime.

Then let's do the same for ForceMode.Impulse:

eIMPULSE = mass * distance /time
eIMPULSE / mass = distance / time
(eIMPULSE / mass) * time = distance

Meaning time is Mathf.Sqrt(Time.fixedDeltaTime) ? Well, as you guessed, on the terrain this doesn't check out.

I like to get the Maths right, how do I get to the result c# code I have shown (that seem correct) starting from the PhysX documentation ?

Answer 1

The confusion you are having is that the PhysX documentation is telling you the units of the values, not the formulas. Forces are given in Newtons whose units are \$kg\cdot m/s^2\$. That is, 1N is the force it takes to accelerate a 1kg mass by 1 m per second each second.