Can You Rearrange Equations to Solve for Unknown Variables in Physics?

[Serj]

In 19 days I will be taking a physics class. I have already taken two algebra classes yet I do not know how to rearrange equations. What are the rules for rearranging an equation. F=m*a, how do you rearrange it so you know what m= ?

 

[HallsofIvy]

You "undo" whatever is done to a. In F= ma, a is multiplied by m. To "undo" that, you do the opposite of "multiply by m"- you divide by m. Dividing both sides of the equation by m, F/m= (ma)/m= a so a= F/m.

 

[quasar987]

mmmh.. by 'algebra' I guess you mean 'arithmetics' ?


The key idea is that F, m and a represent numbers, like 1, 2, 3¼, pi³,...etc. And for any number 'n', there exists another number 'm' such that n*m = 1. This number m is of course, the inverse of n: m = 1/n.

So if for exemple, we want to isolate m in F=ma, we want to multiply both sides by the inverse of a, like so

$$F=ma \Leftrightarrow \frac{1}{a}F=m\frac{a}{a} \Leftrightarrow \frac{F}{a} = m$$

 

[Serj]

You "undo" whatever is done to a. In F= ma, a is multiplied by m. To "undo" that, you do the opposite of "multiply by m"- you divide by m. Dividing both sides of the equation by m, F/m= (ma)/m= a so a= F/m.

t=(Vf-Vi)/a if I wanted to find "a" I would multiply both sides by a, yes? but that would leave me with at=Vf-Vi right? and "a" would not be isolated on one side of the =. what am I doing wrong?

 

[quasar987]

Continue. Divide both sides by t.

 

[TD]

In these type of problems you only need to know two fundamental properties of equations. Equation remain equivalent under two certain operations.


a) An eqaution remains equivalent when you add the same number to both sides (this may be negative if you wish to 'substract').

Example, we want a out of: $$a + b = c \Leftrightarrow a + b - b = c - b \Leftrightarrow a = c - b$$

b) An eqaution remains equivalent when you multiply both sides with the same factor ($$\ne 0$$)

Example, we want a out of: $a\frac{b}{c} = d \Leftrightarrow a\frac{b}{c}\frac{c}{b} = d\frac{c}{b} \Leftrightarrow a = \frac{{dc}}{b}$

That's all you need here

 

[Serj]

thanks everyone

 

[EnumaElish]

b) An eqaution remains equivalent when you multiply both sides with the same factor ($$\ne 0$$)

An eqaution remains equivalent when you multiply both sides with the same factor, including 0.

Generally, an eqaution remains equivalent when you apply an invertible (strictly increasing or strictly decreasing) function to both sides:

a = b implies Log(a) = Log(b);

a = sqrt(b) implies a² = (sqrt(b))² = b.

In general, if a = f(b) and g is the inverse of f, then g(a) = g(f(b)) = b.

 

[TD]

I thought that in equivalent equations, the solutions have to remain the same.
In that case, you have to exclude 0, no?

 

[Serj]

ok I have the equation a=(V2-V1)/t , And I want to make it V2=? . it would be a*V2= (-V1)/t ,is it good so far? but i don't know what to do with "a"

 

[EnumaElish]

ok I have the equation a=(V2-V1)/t , And I want to make it V2=? . it would be a*V2= (-V1)/t ,is it good so far? but i don't know what to do with "a"

You have to pay attention to the parantheses.

a = (V2 - V1)/t
at = V2 - V1
at + V1 = V2

 

[EnumaElish]

I thought that in equivalent equations, the solutions have to remain the same.
In that case, you have to exclude 0, no?

I see what you're saying. I am splitting hairs when I point out that strictly speaking, a = b is preserved under multiplication with zero, which is not covered under your (or, the)definition of equivalent equations.

 

[Serj]

You have to pay attention to the parantheses.

a = (V2 - V1)/t
at = V2 - V1
at + V1 = V2

why did you out everything except V2 on the other side of the = instead of putting V2 were "a" was and put "a" on the other side?

why did you multiply "t" and "a"?
if "t" was the numerator, would you still multiply it with "a"?

 

[ek]

I remember in grade 11 my chem teacher taught us some little triangle method for the equation n=cv to solve for each quantity.

The the next year in Phys12 I sit down at my desk the first day and written on my desk is the little triangle cheat method and someone commented below it "morons use triangles". I always got a kick out of that.

How old are you and what grade are you in? I didn't take a physics class until grade 10 and we were doing stuff like this in grade 7/8. I find it odd that you would be taking a physics class without knowing these algebra fundamentals.

 

[ek]

why did you out everything except V2 on the other side of the = instead of putting V2 were "a" was and put "a" on the other side?

It's the same thing. V = at and at = V are the same thing.

why did you multiply "t" and "a"?

You multiply both sides by t. The t's cancel on the right side and you're left with at = V2 - V1

if "t" was the numerator, would you still multiply it with "a"?

No, then you would divide it out. Divide both sides by t and you're left with a/t = V2-V1 (Which obviously isn't a valid equation)

 

[EnumaElish]

why did you out everything except V2 on the other side of the = instead of putting V2 were "a" was and put "a" on the other side?

why did you multiply "t" and "a"?
if "t" was the numerator, would you still multiply it with "a"?

ek's explanation is right:

a = (V2 - V1)/t
(V2 - V1)/t = a
t(V2 - V1)/t = ta
V2 - V1 = ta
V2 - V1 + V1 = ta + V1
V2 = V1 + ta

 

[Serj]

I've got a problem I don't know how to rearrange. d=Vt+1/2 at^2 ,i'm supposed to find out what t equals.
d/V=Vt/V+1/2 at^2
d/V=t +1/2 at^2
2d/V=t +2*1/2 at^2
2 d/V=t + at^2
(2 d/V)/a=t + at^2/a
(suare root of)(2 d/V)/a=t + (square root of)t^2
(suare root of)(2 d/V)/a=t +t
((suare root of)(2 d/V)/a)/2=2t/2
((suare root of)(2 d/V)/a)/2=t
What did I do wrong? how do I fix it so I am not dividing fractions

 

[Learning Curve]

What do you mean "at power of two"? Do you mean the whole expression is to the power of 2?

And is Vt one variable or two?

 

[VietDao29]

...d/V=Vt/V+1/2 at^2
d/V=t +1/2 at^2

You are wrong at the second line.
You have:
$$a = b + c$$
$$\Leftrightarrow \frac{a}{d} = \frac{b + c}{d} = \frac{b}{d} + \frac{c}{d}$$
We divide both sides by d <> 0.
Anyway, to find t from:
$$d = vt + \frac{1}{2}a t ^ 2$$
$$\Leftrightarrow \frac{a}{2}t ^ 2 + vt - d = 0$$
t is the unknown.
Can you solve:
$$\alpha x ^ 2 + \beta x + \gamma = 0$$
for x?
$$\alpha , \beta , \gamma$$ are already known.
Viet Dao,

 

[Learning Curve]

(ax^2)/a+bx-bx+y-y=(-bx-y)/a

x^2=(-bx-y)/a

square root both sides

x= square root of (-bx-y)/a


That's really messy and I don't know if that's how you would do it.

 

[Moonbear]

I've got a problem I don't know how to rearrange. d=Vt+1/2 at^2 ,i'm supposed to find out what t equals.
d/V=Vt/V+1/2 at^2
d/V=t +1/2 at^2
2d/V=t +2*1/2 at^2
2 d/V=t + at^2
(2 d/V)/a=t + at^2/a
(suare root of)(2 d/V)/a=t + (square root of)t^2
(suare root of)(2 d/V)/a=t +t
((suare root of)(2 d/V)/a)/2=2t/2
((suare root of)(2 d/V)/a)/2=t
What did I do wrong? how do I fix it so I am not dividing fractions

First, you made an error in your first step. If you divide by V, you have to divide EVERYTHING by V, so your second equation would be:
d/V=Vt/V + 1/2(at^2)/V

However, to solve for t in this case, you can't just rearrange the equation. Now you need to go back through some of the other kinematic equations you have learned so far and use those to substitute equivalent terms in this equation so you end up isolating t.

At this point, you need to pay very careful attention that V in this equation is actually Vo, or the initial velocity in case your book uses a different way to symbolize that.

Since this is pretty complicated, I'll get you started, but then you're going to have to look through your notes and see if you can find another equation that helps substitute terms where I leave off. Something else I want to point out from reading what's been done in this thread previously is to remember to pay attention to the order of operations and watch what terms are enclosed in parentheses. In case you've forgotten the order of operations, remember the acronym PEMDAS for Parentheses, Exponents, Multiply, Divide, Add, Subtract. If you aren't careful of doing everything in this order, you'll introduce errors.

1) d=volt+1/2at^2 (your starting equation)
2) d=(Vo + 1/2at)t (factored out t)
3) d=2/2(Vo + 1/2at)t (multiplied the right side of the equation by 2/2; this is not an intuitive step, which is why I wanted to walk you through to this point. You can always multiply by 1 on one side of an equation without changing it's value, and sometimes that means using a fraction like 2/2...back when I was learning this stuff, figuring out when to use that particular trick was really difficult and sometimes it's a matter of staring at the other equations you have to use long enough until one of them jumps out as being a likely candidate if only you had a different denominator somewhere)
4) d=((2Vo + at)/2)t (factored out the 2/2)
5) d= ((Vo + Vo + at)/2)t (substituted Vo + Vo for 2Vo; basic arithmetic)

Okay, now I think this should have gotten you started well on your way. Look through your other equations and see if there's something that allows you to eliminate a term that includes t so you can then rearrange the equation to finish solving for the other t.

 

[HallsofIvy]

$$d= Vt- \frac{1}{2}at^2$$ is a quadratic equation: you can't solve it for t by just "rearranging" the equation. With some specific values of V and a, you might be able to factor but for the general equation you need to use the "quadratic formula". Certainly if you've taken two years of algebra you should know that formula:

If $$ax^2+ bx+ c= 0$$ then $$x= \frac{-b \pm\sqrt{b^2- 4ac}}{2a}$$.

Here $$\frac{a}{2}t^2- Vt+ d= 0$$ so $$t= \frac{V\pm\sqrt{V^2- 2ad}}{a}$$.

 

[Moonbear]

Ack! But of course! D'oh! Serj, I hope my explanation didn't confuse you; it was rather late last night when I responded, and I wound up thinking in an entirely different direction.

Oh, but take note that Halls inverted a sign in his starting equation. It doesn't change how you use the formula, just try your hand at your own starting equation (maybe he did it on purpose so you don't just copy down his answer but have to try to work it out for yourself).

 

[HallsofIvy]

"Oh, but take note that Halls inverted a sign in his starting equation. It doesn't change how you use the formula, just try your hand at your own starting equation (maybe he did it on purpose so you don't just copy down his answer but have to try to work it out for yourself)."

Oh, that's a good excuse! What really happening is, since this is clearly a "motion" problem with acceleration a, I was thinking of a falling object with a= -g and miswrote it.

 

[Moonbear]

Serj, just in case you need your memory refreshed on quadratic equations, here is a site that shows how the final form of the equation was derived so you understand the steps that get you there, not just memorizing the end result.
http://www.purplemath.com/modules/sqrquad.htm

 

[Serj]

I'm finnaly in physics and doing quite well but I am having some trouble.
d=Vi t + 1/2 a t^2 ,t=?
first I would divide both sides by Vi?
d/Vi=t + 1/2 a t^2
then divide both sides by 1/2 a? would it be d/Vi/a or d*a/Vi or d/Vi*a?
i'm going to just guess d/Vi*a (since m/s/s=m/s^2)
d/Vi*a=t +1/2 t^2 would t^2 still be multiplied by 1/2?
(square root of) d/Vi*a= t+1/2 t would the entire left side be square rooted or just the top?
t+1/2 t equals t i believe so t=(sq root of) d/Vi*a is the correct?

Second problem equation:
Vf^2=Vi^2 + 2 ad , d=?
first I would divide both sides by d?
Vf^2/2d=Vi^2 + 2a would it I still multiply d by 2?
here's were I get stuck, what do I do with Vf^2?

 

[VietDao29]

$$d = v_it + \frac{1}{2}at^2$$ And you want to solve for t. Rearrange it as:
$$\frac{1}{2}at^2 + v_it - d = 0$$. And it's a quadratic equation.
A quadratic is defined as:
$$ax^2 + bx + c = 0$$, a, b, c are already known, x is the unknown.
And its solution can be found by letting: $$\Delta = b ^ 2 - 4ac$$
If $$\Delta \geq 0$$, then the equation has solution(s):
$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$
If $$\Delta < 0$$, then the equation has no solution.
You can see http://en.wikipedia.org/wiki/Quadratic_equations for more information.
Here t is unknown.
---------------
For your second question:
$$v_f ^ 2 = v_i^2 + 2ad$$
Isolate d by adding $=v_i^2$ to both sides.
$$v_f ^ 2 - v_i^2 = 2ad$$
To isolate d, divide both sides by 2a:
$$\frac{v_f ^ 2 - v_i^2}{2a} = \frac{2ad}{2a}$$
$$\frac{v_f ^ 2 - v_i^2}{2a} = d$$
Viet Dao,

 

[roger]

$$d = v_it + \frac{1}{2}at^2$$ And you want to solve for t. Rearrange it as:
$$\frac{1}{2}at^2 + v_it - d = 0$$. And it's a quadratic equation.
A quadratic is defined as:
$$ax^2 + bx + c = 0$$, a, b, c are already known, x is the unknown.
And its solution can be found by letting: $$\Delta = b ^ 2 - 4ac$$
If $$\Delta \geq 0$$, then the equation has solution(s):
$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$
If $$\Delta < 0$$, then the equation has no solution.
You can see http://en.wikipedia.org/wiki/Quadratic_equations for more information.
Here t is unknown.
---------------
For your second question:
$$v_f ^ 2 = v_i^2 + 2ad$$
Isolate d by adding $=v_i^2$ to both sides.
$$v_f ^ 2 - v_i^2 = 2ad$$
To isolate d, divide both sides by 2a:
$$\frac{v_f ^ 2 - v_i^2}{2a} = \frac{2ad}{2a}$$
$$\frac{v_f ^ 2 - v_i^2}{2a} = d$$
Viet Dao,

careful with the signs.

 

[VietDao29]

Whoops, stupid typo...
Yes, it should be a '-' sign instead of a '='...
Viet Dao,