Deriving a mechanics equation

[oldoldstudent]

I have a homework assignment, to derive an equation of elastic collisions. In the following equations, m1= mass one, m2 = mass 2, v1 = initial velocity of m1, v2 = initial velocity of m2, v3 = final velocity of m1, and v4 = final velocity of m2. I know from Conservation of Momentum that (m1*v1)+(m2*v2)=(m1*v3)+(m2*v4), and I also know from Conservation of Energy in elastic collisions that [.5m1(v1)^2]+[.5m2(v2)^2]=[.5m1(v3)^2]+[.5m2(v4)^2]. Where I'm trying to get to is the intermediate form v3=[v1(m1-m2)+2*m2v2]/(m1+m2). From there, I can easily get to the final form. And, when I understand how to get to the intermediate form of v3=..., then I can re-do he work for the other part, which is v4=... What I don't know is how to combine these equations (substitution/multiplying/dividing before or after solving for one or another variable?) to get to the intermediate form. Can anybody give me any idea of how to proceed? I spent about 10 hours on this yesterday, both at school and at home. Thanks!

 

[skeeter]

what final result are you trying to derive ?

 

[oldoldstudent]

I'm trying to arrive at v3=[(m1-m2)/(m1+m2)]v1+[2m2/(m1+m2)]v2, and a similar equation but switched around a little for v4. As you can see, getting from the intermediate version posted originally to the above desired final result is fairly trivial. It's the convoluted algebra in the first few steps that has me bamboozled...

 

[Deleted member 4993]

I have a homework assignment, to derive an equation of elastic collisions. In the following equations, m1= mass one, m2 = mass 2, v1 = initial velocity of m1, v2 = initial velocity of m2, v3 = final velocity of m1, and v4 = final velocity of m2. I know from Conservation of Momentum that (m1*v1)+(m2*v2)=(m1*v3)+(m2*v4), and I also know from Conservation of Energy in elastic collisions that [.5m1(v1)^2]+[.5m2(v2)^2]=[.5m1(v3)^2]+[.5m2(v4)^2]. Where I'm trying to get to is the intermediate form v3=[v1(m1-m2)+2*m2v2]/(m1+m2). From there, I can easily get to the final form. And, when I understand how to get to the intermediate form of v3=..., then I can re-do he work for the other part, which is v4=... What I don't know is how to combine these equations (substitution/multiplying/dividing before or after solving for one or another variable?) to get to the intermediate form. Can anybody give me any idea of how to proceed? I spent about 10 hours on this yesterday, both at school and at home. Thanks!

It is just algebra!!! A bit tedious - but simple.

You put up your work - we'll work through to correct your mistake/s, if any present.

 

[mmm4444bot]

There may be an easier approach, but here is what I did.

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EQN 1:

\(\displaystyle \frac{1}{2} \cdot m_1 \cdot v_1^2 \;+\; \frac{1}{2} \cdot m_2 \cdot v_2^2 \;=\; \frac{1}{2} \cdot m_1 \cdot v_3^2 \;+\; \frac{1}{2} \cdot m_2 \cdot v_4^2\)

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EQN 2:

\(\displaystyle m_1 \cdot v_1 + m_2 \cdot v_2 \;=\; m_1 \cdot v_3 + m_2 \cdot v_4\)

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FACTOR AND SIMPLIFY EQN 1 TO OBTAIN EQN 3:

\(\displaystyle m_1 \cdot (v_1 + v_3) \cdot (v_1 - v_3) \;=\; m_2 \cdot (v_4 + v_2) \cdot (v_4 - v_2)\)

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FACTOR EQN 2 AND SOLVE FOR v[sub:6w5f83ug]4[/sub:6w5f83ug] - v[sub:6w5f83ug]2[/sub:6w5f83ug] TO OBTAIN EQN 4:

\(\displaystyle v_4 - v_2 \;=\; \frac{m_1}{m_2} \cdot (v_1 - v_3)\)

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SUBSTITUTE EXPRESSION FOR v[sub:6w5f83ug]4[/sub:6w5f83ug] - v[sub:6w5f83ug]2[/sub:6w5f83ug] INTO EQN 3 AND SOLVE FOR v[sub:6w5f83ug]4[/sub:6w5f83ug] TO OBTAIN EQN 5:

\(\displaystyle v_4 \;=\; v_1 - v_2 + v_3\)

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SUBSTITUTE EXPRESSION FOR v[sub:6w5f83ug]4[/sub:6w5f83ug] INTO EQN 2 AND SOLVE FOR v[sub:6w5f83ug]3[/sub:6w5f83ug] TO OBTAIN EQN 6:

\(\displaystyle v_3 \;=\; \frac{v_1 \cdot (m_1 - m_2) + 2 \cdot m_2 \cdot v_2}{m_1 + m_2}\)

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If you spent 10 hours on this, then you might have been going in circles. (I've been there.)

Cheers,

~ Mark

 

[oldoldstudent]

Holy smokes, you've done it!!! I would never ever have thought to solve for something as weird as (v4-v2)...Although I solved for and plugged back in every possible other thing, I think. I and a classmate have been working on this for about 40 man-hours. I am eternally grateful.

 

[mmm4444bot]

... I would never ever have thought to solve for something as weird as (v4-v2) ...

I did not start out thinking about finding an expression for v[sub:4u89jka1]4[/sub:4u89jka1] - v[sub:4u89jka1]2[/sub:4u89jka1]. I began by simplifying both equations to see how I might eliminate the squared terms. After factoring the difference of squares, I then looked at EQN 2 to see how I might eliminate the v[sub:4u89jka1]4[/sub:4u89jka1] term.

The sooner you begin to look at expressions as normal, versus weird, the sooner you will be open to seeing various substitutions and eliminational approaches. (There is nothing weird about the expression v[sub:4u89jka1]4[/sub:4u89jka1] - v[sub:4u89jka1]2[/sub:4u89jka1], although there may be something weird about the word "eliminational"!)

Cheers,

~ Mark