# Trying to avoid the quartic formula

#### CRYS1SX18

I need help solving for t in the following equation,

$$|P + Vt + A(t^2)/2| = st$$

where P V and A are vectors and s and t are scalars.

I've tried squaring both sides which results in the equation

$$(|A|^2/4)t^4+V.Gt^3+(P.G+|V|^2)t^2+2P.Vt + |P|^2 = s^2t^2$$

$$(|A|^2/4)t^4+V.Gt^3+(P.G+|V|^2-s^2)t^2+2P.Vt+|P|^2=0$$

The only way I can think to solve this is through the quartic formula which is too unwieldy for my purposes and was wondering if anyone had any ideas.

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#### jedishrfu

Mentor
Welcome to PF!

This looks like homework and if so you should post it under the correct topic and use the homework template where you describe the problem, show relevant formula and your attempt at a solution.

Also it might help if you described where this equation is being used and what the P, V, A and s, t refer to.

Also your solution introduces a G which looks to be a vector too.

#### CRYS1SX18

It isn't homework.

The problem is to hit a target with constant acceleration with a projectile that travels at constant speed.

Known variables are:

s(scalar) - the initial speed of your projectile

P(vector) - the initial position of the target relative to the shooter

V(vector) - the initial velocity of the target relative to the shooter

A(vector) - the constant acceleration of the target relative to the shooter

This means the targets equation of motion is
P(t) = P + Vt + 0.5 A t^2 (this formula does work in 3d)

where t is time

and no matter what direction you launch your projectile in it will always be s * t away from the shooter

and that's where |P + Vt+(At^2)/2| = st comes from

once that's solved for t you can plug it back into the equation of motion and aim at where the target will be at that time but I'm not sure how I would go about doing that without resorting to the quartic formula

I realize this is a physics problem but I've already done the physics aspect of it and just need help solving the mathematical equations.

also G is supposed to be A sorry about that

Last edited:

#### jedishrfu

Mentor
How about a numeric solution? You could use freemat a free version of MATLAB to compute the answer.

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