Shooting basic projectile under constant speed and various angles and then joining points of where individual projectile gets highest, that is at the apex of parabola, you get an ellipse. (I saw this at 1ucasvb recently.) Introducing air resistance makes force gain the additional air drag term -b * r'(t) with r(t) the coordinate vector and gets the ballistic ellipse at b = 0 squashed to a semi circle.

Now, I don’t know about you but me I am fascinated that everything, whether the ellipse focus, the angle that shoots a trajectory through that focus or the curve under air resistance, everything can be derived symbolically in closed form. I never really excelled at math yet I marvel at equations. I just stare at them sometimes.

Shooting basic projectile under constant speed and various angles and then joining points of where individual projectile gets highest, that is at the apex of parabola, you get an ellipse. (I saw this at 1ucasvb recently.) Introducing air resistance makes force gain the additional air drag term -b * r'(t) with r(t) the coordinate vector and gets the ballistic ellipse at b = 0 squashed to a semi circle.

Now, I don’t know about you but me I am fascinated that everything, whether the ellipse focus, the angle that shoots a trajectory through that focus or the curve under air resistance, everything can be derived symbolically in closed form. I never really excelled at math yet I marvel at equations. I just stare at them sometimes.

Notes

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