Trouble Simplifying/Factorising

**dkaksl** · December 24th 2009, 07:56 PM

I have a rather straightforward Physics problem to solve, only I can't for the life of me simplify the answer to what it needs to be. What am I doing wrong?

$<br /> v_1=\frac{m}{m+M}v_o<br />$

$\frac{1}{2}mv_o^2=\frac{1}{2}mv_1^2+\frac{1}{2}Mv_ 1^2+\frac{1}{2}kd^2$

The answer is in the form $d=$ and it's a simplification of the information I provided here in the post.

I get as far as $d=\sqrt{\frac{1}{k}(mv_o^2-(m+M)v_1^2)}$

The book plugs in $v_1$

and simplifies to $d=\sqrt{\frac{mM}{k(m+M)}}\times{v_o}$

How?

**Wilmer** · December 24th 2009, 08:50 PM

Originally Posted by dkaksl

$<br /> v_1=\frac{m}{m+M}v_o<br />$

$\frac{1}{2}mv_o^2=\frac{1}{2}mv_1^2+\frac{1}{2}Mv_ 1^2+\frac{1}{2}kd^2$

Changing your v1 to v and v0 to w, then your 2 equations are:

v = mw / (m + M) [1]

mw^2 = mv^2 + Mv^2 + kd^2 [2]

Simplifying:
m + M = mw / v [1]
mw^2 - kd^2 = v^2(m + M) [2]

Substitute [1] in [2] ; OK?

**dkaksl** · December 24th 2009, 08:57 PM

Very useful, easy-to-understand solution. Thanks a lot.

**dkaksl** · December 25th 2009, 12:13 AM

Hello again.

Just got to checking the textbook and unfortunately the question was to define $d$ in the terms $m, M, k$ and $v_o$ .

Cancelling out $(m + M)$ got me an answer with $v_1$ .

Edit:

$kd^2=mv_o^2-(m+M)v_1^2$

if $v_1=\frac{mv_o}{m+M}$ then what is $v_1^2$ ?

$\frac{mv_o}{m+M}\times\frac{mv_o}{m+M}$ which is

$\frac{m^2v_o^2}{(m+M)^2}$

$kd^2=mv_o^2-(m+M)\times\frac{m^2v_o^2}{(m+M)^2}$

$kd^2=mv_o^2-\frac{m^2v_o^2}{(m+M)}$

$kd^2=\frac{mv_o^2(m+M)}{(m+M)}-\frac{m^2v_o^2}{(m+M)}$

$kd^2=\frac{mv_o^2(m+M)-m^2v_o^2}{(m+M)}$

$kd^2=\frac{m^2v_o^2+mMv_o^2-m^2v_o^2}{(m+M)}=\frac{mM}{(m+M)}\times{v_o^2}$

$d=\sqrt{\frac{1}{k}\times{\frac{mM}{(m+M)}}}\times {v_o}$

Last edit: Thanks a lot. I see where I made a mistake in expanding. Learned a lot today. Merry Christmas.

**Wilmer** · December 25th 2009, 04:09 AM

Originally Posted by dkaksl

Just got to checking the textbook and unfortunately the question was to define $d$ in the terms $m, M, k$ and $v_o$ .

Ahhh; well then, we need to get rid of your v1 (my v); the 2 equations:

v = mw / (m + M) [1]

mw^2 = mv^2 + Mv^2 + kd^2 [2]

square [1]: v^2 = m^2w^2 / (m + M)^2
rearrange [2]: v^2 = (mw^2 - kd^2) / (m + M)

m^2w^2 / (m + M)^2 = (mw^2 - kd^2) / (m + M)

m^2w^2 / (m + M) = mw^2 - kd^2

m^2w^2 = mw^2(m + M) - kd^2(m + M)

kd^2(m + M) = mw^2(m + M) - m^2w^2

kd^2(m + M) = mw^2(m + M - m)

kd^2(m + M) = mMw^2

d^2 = mMw^2 / (k(m + M))

d = wSQRT[mM / (k(m + M))] ........Merry Xmas!

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