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.

 

[jedishrfu]

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.

Please be aware that PF will help you solve it but won't give you the answer without some work on your part.

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

 

[jedishrfu]

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

 

[blue_raver22]

I understand the need to avoid using complex and unwieldy formulas in order to solve equations efficiently. In this case, instead of using the quartic formula, we can try to simplify the equation by using some mathematical techniques.

Firstly, we can try to rearrange the equation to isolate the variable t on one side. We can do this by subtracting st from both sides of the equation, which gives us:

|P + Vt + A(t^2)/2| - st = 0

Next, we can try to expand the absolute value term by using the definition of absolute value. This will give us two separate equations, one for the positive case and one for the negative case:

P + Vt + A(t^2)/2 - st = 0 (for P + Vt + A(t^2)/2 > 0)

and

-(P + Vt + A(t^2)/2) - st = 0 (for P + Vt + A(t^2)/2 < 0)

We can then simplify these equations by using the properties of vectors and scalars. For example, we can factor out t from the first equation, giving us:

t(A/2 + V + s) + P = 0

We can also use the dot product to simplify the equations further. For instance, we can rewrite the equation as:

t(A.G/2 + V.G + s) + P.G = 0

Using these techniques, we can simplify the equation to a quadratic form, which can be solved using the quadratic formula. This approach may be more efficient and less complicated than using the quartic formula.

Additionally, we can also try to use numerical methods, such as Newton's method or the secant method, to approximate the solution. These methods involve iteratively improving an initial guess for the value of t until we reach a satisfactory level of accuracy.

In conclusion, while the quartic formula may be a valid method for solving this equation, there are alternative approaches that may be more efficient and less cumbersome. By using mathematical techniques and numerical methods, we can simplify the equation and find a solution without having to resort to using the quartic formula.