get Projectile hit moving target

calthabis

Hello i need this för a coding project im currently occupying myself with =)

ok so the system is in a 2D space, the projectile starts at a height H and is supossed to hit target T. the targets y.coord is constant Ty but his x has a speed Tv towards the projectiles start. the target starts at a distance D from the projectile. the projectile has a given starting speed of Pv0 and the angle A.

oh right and gravity is affecting the projectile =)


SO! what i need is a formula that can tell what angle I need to hit the target.

Thanks in advance

calthabis

Was this too difficult or is there not enough information??

DaleSpam

You can write an expression for the location of the projectile as a function of t and the angle, and you can write an expression for the location of the target as a function of t. Set them equal to each other and you have an equation describing when they collide. That leaves 2 equations in 2 unknowns so you can solve for t and the angle.

calthabis

This is how far ive got. Im stuck so can you please say how i can solve the rest or if there is some other way to solve it.
it might be quite hard to follow on screen so i suggest you follow my calculations on paper.

the target's y coordinate is always 0

Tv = the target's X velocity (speed)
Pv = the projectile's velocity

d0 = the target's x coordinate at t=0
y0 = the projectile's y coordinate at t=0


Pvx = Pv * cosa
Pvy = Pv * sina +gt

Px = Pv *cosa *t
Tx = d0 + Tv *t
Py = y0 + Pv* sina + (g*t^2) / 2
Ty = 0


Tx = Px
Ty = Py


(1.) d0 + Tv *t = Pv *cosa *t
(2.) 0 = y0 + Pv* sina + (g*t^2) / 2


(1.) cosa^2 = ( ( d0 + Tv *t )/(Pv * t) )^2
(2.) sina^2 = (( -y0 -(g*t^2) / 2 ) / Pv)^2


cosa^2 + sina^2 = 1
(1.) + (2.) = 1

( ( d0 + Tv *t )/(Pv * t) )^2 + (( -y0 -(g*t^2) / 2 ) / Pv)^2 = 1

DaleSpam

You shouldn't try to solve this by hand. You should either plug the equations into a symbolic math package, like Mathematica, or you should solve it numerically. Just for grins I used Mathematica. I had these two equations describing repectively the x and y coordinates of the collision:

T0x + t Tvx == Pv t Cos[a]
2 P0y + g t^2 + 2 Pv t Sin[a] == 0

Solving this simple appearing system for t and a resulted in 8 roots, each one of which was over 100 pages of output.

You could probably solve this a more clever way to get fewer roots and slightly simpler expressions. One typical way is to not solve for the angle but rather the x and y components. But you would never get something truly simple. If you are coding I would recommend using a numerical solver. There are lots of packages available, many for free.