CANNON PARTICLE !!!

STARTER ANGLE:

INITIAL VELOCITY:

GRAVITY

FADE FACTOR

so yeah once in a time i was kinda bored to i decided to make a cannon program in the about:blank page using only javascript, it worked and now i want to share it with yall in this new revamped version, to use just:

1. select your prefered initial velocity(positive), angle(from 0 to 90 degrees) and gravity
2.click the FIRE button
3. optional: with the cancel button u can stop the bullet at a moment

have fun with it!

you can also find this animation-less version, but be careful as to not make distance of travel too far!

how does this work????

quite simple, the process for how i made this thingy was kinda like this:

we have a particle with starting velocity:

(v_x, v_y)

now, at a moment in time t , the velocity of the particle will be:

v(t) = (v_x, v_y - gt)

by doing an integral to get the position we obtain:

∫ v(t) dt = ∫ (v_x, v_y - gt) dt

p(t) = (t v_x, t v_y - g t ² 2 )

now, how can we make this a function? well, notice how the x conmponent is on a linear form t v_x , so if we define t' = t v_x , then:

p(t') =(t', v_y v_x t - g2 v_x ² t ² )

look at that! it has got the structure of a function (x, f(x)), with this we got a function that we can evaluate:

f(x) = v_y v_x x - g2 v_x ² x ²

this function is now then evaluated from x=0 to the value where our particle will hit the floor

and how u got the data??

quite simple, we just need to solve three equations:

TRAVEL TIME

for travel time, we solve this equation:

v_y - gt = -v_y

which with some manipulations:

2v_y = gt

t = 2 v_y g

this is the time it takes for particle to hit ground again

DISTANCE OF TRAVEL

we just take the travel time, and multiply by horizontal component of velocity:

d = 2 v_x v_y g

HIGHEST ALTITUDE:

get distance of travel, divide by two, and the evaluate the result in the function constructed before:

d/2 = v_x v_y g

h = v_y v_x * v_x v_y g - g2 v_x ² ( v_x v_y g ) ²

with some cancelations, we obtain:

h = v_y² g - v_y² 2g

h = v_y² 2g