Calculating the result of adding two velocities

Question

I have a simple 2D system in which a velocity is represented as a speed (some value) and direction (an angle). If I have an object travelling at a given velocity v, when another velocity v₂ acts upon it I want to calculate the resultant velocity v₃.

I have created the following illustration to explain the physics system I wish to implement. The speeds and angles may not be exactly correct because I do not know how to calculate them yet - that is the crux of this question - yet they serve as a fairly close guideline.

Is there an elegant solution for calculating the resultant velocities v₃, when new velocity v₂ acts upon an object travelling at v?

Please note that I have a very limited understanding of mathematics but a good understanding of programming. I am coding in TypeScript but any answer in JavaScript, pseudo code, or very clear explanation (not riddled with cryptic mathematical symbols) is helpful.

Answer 1

The simplest solution consists in converting to cartesian coordinates, add the coordinates there, then convert back to polar coordinates.

converting to cartesian:

x = speed * cos(direction)
y = speed * sin(direction)

back to polar:

speed = sqrt(x * x + y * y)
direction = atan2(y, x)

It might be a better idea to store your velocities in cartesian coordinates.

Edit: using your variable names: v, v2, v3, and angles a, a2, a3:

x = v * cos(a)
y = v * sin(a)

x2 = v2 * cos(a2)
y2 = v2 * sin(a2)

x3 = x + x2
y3 = y + y2

v3 = sqrt(x3 * x3 + y3 * y3)
a3 = atan2(y3, x3)

Answer 2

So you have an object flying in the direction v1. In order to support velocities, you just have to add them to v1 for each frame / simulation step:

var v2;
while(simulation is running){
    v1 = v1 + v2;
}

You can represent v1, v2 as vectors with the components x and y.

In order to convert you values, you can use x = speed * cos(direction) and y = speed * sin(direction) as supposed by Rémi.

Your addition will then look as follows:

v1.x = v1.x + v2.x;
v1.y = v1.y + v2.y;

In order to get the angle of a vector, you can use the formula tan(angle) = y / x, resulting in

angle = atan(y/x)

The according JavaScript call is var angle = atan2(v1.y, v1.x);.

The length of your vector (= speed of you object) can be calculated with

speed = sqrt(v1.x * v1.x + v1.y * v1.y)