How to apply 2D gravitational effects to sprites

Question

Within a 2D infinite space, I have two circular sprites that represent massive bodies in outer space.

Each body has a pair of x , y coordinates, a mass , a speed and a direction (in radians).

On each frame of animation, I'm running the following code for each body (where this is the body being updated, and other is the other body):

x, y = other.x - this.x, other.y - this.y

angle = atan2(y, x)
distance = root(square(x) + square(y))
force = this.mass * other.mass / square(distance)

Note: I'm ignoring G as it's just a multiplier.

I know how to move the bodies around, based on their coordinates, speed and direction, but do not know how to update this.speed and this.direction to simulate gravity.

Answer 1

The gravitational force acting on a given body is represented as a vector and produces acceleration with components ( ax and ay ), which are calculated (based on what you already have) like this:

squared_distance = square(x) + square(y)
distance = sqrt(squared_distance)

accel = other.mass / squared_distance    
ax = accel * x / distance 
ay = accel * y / distance

Note that the angle (force/acceleration direction) is not needed.

Each body should have an associated velocity (instead of speed ), which should be a two-component vector ( vx and vy ). It is updated like this (where dt is the time interval between updates):

this.vx += ax * dt
this.vy += ay * dt

Once the velocity of a given body has been updated, it can then be repositioned (updating its x , y coordinates) like this:

this.x += this.vx * dt
this.y += this.vy * dt

You can calculate the speed and direction if you need them, but they are not needed here.