Is this equation solvable for v?

• B
Hello
I am attempting to solve this equation (a physics project); however, I seem to be getting stuck in a cycle between attempting to undo the square root and then dealing with the resulting quadratic. I have run out of creative solutions to get the variable v by itself. Any help would be greatly appreciated.

F = (m/(1-v2/c2)-1/2) * (v/t-b/t)

Answers and Replies

By squaring both sides I can isolate a quadratic equation in v but it's huge.... Did you try that?

mathman
Science Advisor
As written the equation looks funny. You have an expression raised to power -1/2 in the denominator. Why not +1/2 in the numerator?

As written the equation looks funny. You have an expression raised to power -1/2 in the denominator. Why not +1/2 in the numerator?
Well you could rearrange the equation to be

F * (1-v^2/c^2)^1/2 = m * (v/t - b/t)
but i do not understand how that will really solve anything. If I now square both sides to get rid of square root over (1-v^2/c^2) I now end up with a v^2 -2vb + b^2 on the other side which is an issue because now how do I get the v byitself over here without messing up the other side again. That is the issue I have been happening. by the way, is there a better way to write these equations. I feel as though the way I have been typing this equation in text format may be a bit confusing?

PeroK
Science Advisor
Homework Helper
Gold Member
2020 Award
Well you could rearrange the equation to be

F * (1-v^2/c^2)^1/2 = m * (v/t - b/t)
but i do not understand how that will really solve anything. If I now square both sides to get rid of square root over (1-v^2/c^2) I now end up with a v^2 -2vb + b^2 on the other side which is an issue because now how do I get the v byitself over here without messing up the other side again. That is the issue I have been happening. by the way, is there a better way to write these equations. I feel as though the way I have been typing this equation in text format may be a bit confusing?

What's the problem? If you square both sides you get a quadratic in ##v##. The coefficients are a bit messy, but there's nothing you can do about that.

Well you could rearrange the equation to be

F * (1-v^2/c^2)^1/2 = m * (v/t - b/t)
but i do not understand how that will really solve anything. If I now square both sides to get rid of square root over (1-v^2/c^2) I now end up with a v^2 -2vb + b^2 on the other side which is an issue because now how do I get the v byitself over here without messing up the other side again. That is the issue I have been happening. by the way, is there a better way to write these equations. I feel as though the way I have been typing this equation in text format may be a bit confusing?

You can use LaTeX to write the equations. Using LaTeX your equation is $$F = \frac{m (\frac{v}{t} - \frac{b}{t})}{\sqrt(1 - \frac{v^2}{c^2})}$$ A guide to LaTeX can be found here and here.

Regarding your equation. Squaring it you get: $$F^2 = \frac{m^2 (\frac{v^2}{t^2} + \frac{b^2}{t^2} - 2\frac{vb}{t^2})}{(1 - \frac{v^2}{c^2})}$$

Algebraically manipulating should get you a quadratic in v.

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You can use LaTeX to write the equations. Using LaTeX your equation is $$F = \frac{m (\frac{v}{t} - \frac{b}{t})}{\sqrt(1 - \frac{v^2}{c^2})}$$ A guide to LaTeX can be found here and here.

Regarding your equation. Squaring it you get: $$F^2 = \frac{m^2 (\frac{v^2}{t^2} + \frac{b^2}{t^2} - 2\frac{vb}{t^2})}{(1 - \frac{v^2}{c^2})}$$

Algebraically manipulating should get you a quadratic in v.

Okay, thanks for the LaTex guide. Now could you show me how you would algebraically solve it from where you stopped? I still end up getting stuck.

mathman
Science Advisor
Okay, thanks for the LaTex guide. Now could you show me how you would algebraically solve it from where you stopped? I still end up getting stuck.
Write out the equation at the point you are stuck. To me it is a simple quadratic.

Okay, thanks for the LaTex guide. Now could you show me how you would algebraically solve it from where you stopped? I still end up getting stuck.

That would be solving the equation for you. Like @mathman said why don't you show us where you're getting stuck?! And we'll help you out.

That would be solving the equation for you. Like @mathman said why don't you show us where you're getting stuck?! And we'll help you out.
Okay, no problem. I'll bold the variable v.

First I multiply both sides by the denominator 1- v^2/c^2 and distribute the m^2 to get

F^2 -(F^2*v^2)/c^2 = (m^2*v^2)/t^2 + (m^2*b^2)/t^2 - (2*v*b*m^2)/t^2

Now I put all the terms with v on one side.

F^2 -(m^2*b^2)/t^2 = (m^2*v^2)/t^2 + (F^2*v^2)/c^2 - (2*v*b*m^2)/t^2

This is were I get stuck. I can't pull out any like terms , since not all the v are squared I can't pull them out either. This is the point where I'm stuck. I'm probably just overlooking something but I would appreciate it if you could now show me where my error is.

PeroK
Science Advisor
Homework Helper
Gold Member
2020 Award
Okay, no problem. I'll bold the variable v.

First I multiply both sides by the denominator 1- v^2/c^2 and distribute the m^2 to get

F^2 -(F^2*v^2)/c^2 = (m^2*v^2)/t^2 + (m^2*b^2)/t^2 - (2*v*b*m^2)/t^2

Now I put all the terms with v on one side.

F^2 -(m^2*b^2)/t^2 = (m^2*v^2)/t^2 + (F^2*v^2)/c^2 - (2*v*b*m^2)/t^2

This is were I get stuck. I can't pull out any like terms , since not all the v are squared I can't pull them out either. This is the point where I'm stuck. I'm probably just overlooking something but I would appreciate it if you could now show me where my error is.

I thought you were aiming at a quadratic? A quadratic in ##v## has terms in both ##v## and ##v^2## plus a constant term. That's what you've got, isn't it? All you have to do is put the two ##v^2## terms together.

What you have is of the form:

##c = a_1 v^2 + a_2 v^2 - bv##

And that's a quadratic.

Mastermind01
I thought you were aiming at a quadratic? A quadratic in ##v## has terms in both ##v## and ##v^2## plus a constant term. That's what you've got, isn't it? All you have to do is put the two ##v^2## terms together.

What you have is of the form:

##c = a_1 v^2 + a_2 v^2 - bv##

And that's a quadratic.
I was so bent on finding a direct equation I completely missed that solution. Thanks for all the help.