Solve for v: "How to Solve for v with Theta, g, and x

[marthkiki]

Homework Statement

theta = tan^-1 {{v^2 - [v^4-g(gx^2)]}^1/2}/gx
I know that x is 30, theta is 45, g is 9.81 m/sec^2.
I'm trying to solve for v.

Homework Equations

The Attempt at a Solution

I've gotten all the way down to
v^2 = (v^4-86612.49)^1/2 + 4.165
What do I do from here?

 

[rock.freak667]

The Attempt at a Solution

I've gotten all the way down to
v^2 = (v^4-86612.49)^1/2 + 4.165
What do I do from here?

you'll need to re-write it as


v² - 4.165= (v⁴-86612.49)^1/2

then square both sides and solve. You may need to use a numerical method. You'll need to test your solutions as squaring both sides might give you false answers.

 

[Jebus_Chris]

Once you square it you will be left with only 1 V term. It may be easier to replace your constants with variables. Then just substitute them in once you find V.

 

[HallsofIvy]

How "only 1 V term"? There will be $v^4$ and $v^2$ terms but then you can let $u= v^2$ and will have a quadratic to solve for u.

 

[marthkiki]

you'll need to re-write it asv² - 4.165= (v⁴-86612.49)^1/2

then square both sides and solve. You may need to use a numerical method. You'll need to test your solutions as squaring both sides might give you false answers.

By squaring both sides, I get v^2 - 17.35 = v^4 - 86612.49.
That gets me -17.35 = -86612.49.
Is this what you mean by a false answer?

 

[marthkiki]

How "only 1 V term"? There will be $v^4$ and $v^2$ terms but then you can let $u= v^2$ and will have a quadratic to solve for u.

If i do that, I end up with u = v² + 416.2
then the discriminant will end up negative leaving me with an imaginary velocity. Did I do this correctly?

 

[marthkiki]

If i do that, I end up with x = 86612.49 and y = 4.165. I then end up with v^4 - y^2 = v^4 - x, which again gets me to -y^2=x.

 

[Jebus_Chris]

v^2 -4.165 = (v^4-86612.49)^1/2
$$v^2 - x = (v^4-y)^{1/2}$$
$$v^4 - 2v^2x + x^2=v^4-y$$
$$-2v^2x + x^2=-y$$
$$v=\sqrt{\frac{x^2+y}{2x}}$$
there u go

 

[slider142]

By squaring both sides, I get v^2 - 17.35 = v^4 - 86612.49.

The left-hand side is not squared properly. The left hand side should be $(v^2 - 17.35)^2 = v^4 - 2(17.35)v^2 + (17.35)^2$. The v⁴'s will then cancel and you will be left with a quadratic in v. You will need to check the two answers you get, as squaring both sides of an equation introduces new solutions that may not solve the original equation. Ie., x = -1 and x² = 1, which introduces the solution x = 1.

 

[marthkiki]

Thank you so much!
I get it now. I totally squared the left wrong.