How Can Algebra Prove the Elastic Collision Equation?

[Shadowsol]

1.

I was given the equation: (Va1-Vb1)=-(Vb2-Va1) I am suppose to prove this equation with algrebra as to why this is true. I can only use algebra, and can't use any numbers.

This is an equation for elastic collisions. There is an object;A and an Object;B. My teacher said the problem would take a page or more of algrebra to prove it. Anyone have any idea how to start the problem?

Any help is appricated.

Homework Equations

We did a lab when we were given this problem, and the lab included the formula for kinetic energy. The lab showed in elastic collisions, momentum and Kinetic energy is conserved.



3. I honestly have no idea where to start.

 

[Doc Al]

What's conserved in an elastic collision?

 

[Shadowsol]

Kinetic energy and momentum. I tried it again and I'm still lost

 

[Doc Al]

Kinetic energy and momentum. I tried it again and I'm still lost

That's what you need. Hint: Start by writing both equations. Rearrange each so that all the Va terms are on one side; Vb terms on the other.

Then do a bit of algebra.

 

[Shadowsol]

So write out the Momentum equation and the Kinetic energy equation and set them equal to each other? And the final result of that should be the equation that I first posted right?

 

[Doc Al]

You're not going to set them equal to each other (not even sure what that means!), but you will combine them. The first step is to rearrange each equation (momentum and KE) as I suggested in the last post.

 

[Shadowsol]

For the kinetic energy equation, there is only one m and v term. Is the m and v term in the Kinetic energy equation the v1 and m1 or v2 and m2?

 

[Shadowsol]

Ok I set them as this-
FT+m1v1=m2v2
.5m1v2squared=.5m2v2squared

Im hoping the Kinetic Energy equation is setup right. The masses cancel out in both equations. But I can't seem to get rid of FT. Also, do I combine them and than solve to get the original equation I was given?

 

[Doc Al]

Your conservation of momentum equation should look something like this:

m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'

Your conservation of energy equation should look something like this:

1/2m_1v_1^2 + 1/2m_2v_2^2 = 1/2m_1v_1'^2 + 1/2m_2v_2'^2

That's your starting point. As a first step towards combining these equations, I recommend that you rewrite each equation, putting the terms relating to m1 on the left and the terms relating to m2 on the write.

Do that and we'll see what's next.

 

[Shadowsol]

Ok for the initial equation, when I moved stuff around, I found that vb1=vb2. The final result I got was vb1(va1-vb1)=-vb2(vb2-va2). The vb1 and vb2 part is proved from what I first said, and what's left is the orginal equation.

Can anyone see if this makes sense?

 

[Shadowsol]

Ok, i'll try and do what you just posted to see how that works out

 

[Shadowsol]

Ok, I got:

ma(Va1squared-va2squared)=mb(vb2squared-vb1squared)

I got all the ma and mb terms to their respective sides.

Now do I combine that with the KE equation? Would the Ke equation be:
.5ma1va1=.5mb1vb2 ?

Since object A's intial velocity/momentum has Kinetic energy, but when it hits object b, its Kinetic energy gets transferred to object b, so object b's intial would have no Kinetic energy, but when it gets hit, it does?

 

[Doc Al]

Ok, I got:

ma(Va1squared-va2squared)=mb(vb2squared-vb1squared)

I got all the ma and mb terms to their respective sides.

Good. That's the rearranged KE equation. Now do the same for the momentum equation.

 

[Shadowsol]

Ok after trying it your way, it seems I may have copied the original equation wrong.

Is it va1-vb1=-vb2-va2 as opposed to the one I first posted, where it is va1 on the right side instead of va2?

The va1 on the left and right side doesn't seem to make sense, so I think I posted the wrong intial equation.

 

[Doc Al]

Yes, your original equation had a typo in it. (I had meant to point that out.)

 

[Shadowsol]

Is there a paranthesis on the right side?

 

[Shadowsol]

I keep getting va1-vb1=vb2-va2, I can't seem to get vb2 to be negative.

 

[Doc Al]

Is there a paranthesis on the right side?

The equation you want to prove, using your notation, is:
va1-vb1 = vb2-va2 = - (va2 - vb2)

Note that va1 - vb1 is the relative velocity of one mass with respect to the other. The equation you are trying to prove states that the relative velocity reverses in an elastic collision.

 

[Doc Al]

I keep getting va1-vb1=vb2-va2,

That's the one you want! If you've gotten that far, you're done.

I can't seem to get vb2 to be negative.

Good thing!

 

[Shadowsol]

Thank you so much for your help! :)