simulate gravity in a 2D game

What part are you stuck at?

I keep a separate position and velocity pair, and pass that into my acceleration function. Then when I'm done adjusting the position and velocity, I call game_object.transform.position = temp_position; -- I do not use the transform.velocity variable.

Here is the Runge-$$anonymous$$utta order 4 integrator:

 void proceed_rk4(ref tennis_parameters t, ref Vector3 pos, ref Vector3 vel, Vector3 ang_vel, float dt)
 {
     const float one_sixth = 1.0f / 6.0f;
     
     Vector3 k1_velocity = vel;
     Vector3 k1_acceleration = acceleration(ref t, pos, k1_velocity, ang_vel);
     Vector3 k2_velocity = vel + k1_acceleration * dt * 0.5f;
     Vector3 k2_acceleration = acceleration(ref t, pos + k1_velocity * dt * 0.5f, k2_velocity, ang_vel);
     Vector3 k3_velocity = vel + k2_acceleration * dt * 0.5f;
     Vector3 k3_acceleration = acceleration(ref t, pos + k2_velocity * dt * 0.5f, k3_velocity, ang_vel);
     Vector3 k4_velocity = vel + k3_acceleration * dt;
     Vector3 k4_acceleration = acceleration(ref t, pos + k3_velocity * dt, k4_velocity, ang_vel);
     
     vel += (k1_acceleration + (k2_acceleration + k3_acceleration) * 2.0f + k4_acceleration) * one_sixth * dt;
     pos += (k1_velocity + (k2_velocity + k3_velocity) * 2.0f + k4_velocity) * one_sixth * dt;
 }

And here is the acceleration function:

 Vector3 acceleration(ref tennis_parameters t, Vector3 pos, Vector3 vel, Vector3 ang_vel)
 {
     // $$anonymous$$agnus effect, in metres per second, per second
     // http://farside.ph.utexas.edu/$$anonymous$$ching/329/lectures/node43.html
     // http://spiff.rit.edu/richmond/baseball/traj/traj.html
     Vector3 magnus_accel = Vector3.Cross(vel, ang_vel);
     magnus_accel *= 0.5f * t.air_density * t.lift_coeff * t.ball_cross_section / t.ball_mass;
     
     // Wind and drag, in metres per second, per second
     Vector3 drag_vel = t.wind_velocity - vel;
     Vector3 drag_accel = drag_vel * drag_vel.magnitude * 0.5f * t.air_density * t.drag_coeff * t.ball_cross_section / t.ball_mass;
     
     return t.gravitational_acceleration + magnus_accel + drag_accel;
 }

I forgot to mention:

 gravitational_acceleration = new Vector3(0, -9.81f, 0);