Collisions- Simple Algebra Problem

[verd]

Collisions-- Simple Algebra Problem...

Hi,

So... I think I'm having a simple algebra problem-- I was just wondering if someone could point out my error. This is the problem:

Block 1, of mass m_{1}, moves across a frictionless surface with speed u_{i}. It collides elastically with block 2, of mass m_{2}, which is at rest (v_{i}=0). After the collision, block 1 moves with speed u_{f}, while block 2 moves with speed v_{f}. Assume that m_{1} > m_{2}, so that after the collision, the two objects move off in the direction of the first object before the collision.

[Image]

What is the final speed u_{f} of block 1?


So-- I find m_{2}v_{f} using the law of conservation of momentum:
m_{2}v_{f} =m_{1}u_{i}-m_{1}u_{f}

And I find m_{2}v_{f}^2 using the law of conservation of kinetic energy:
m_{2}v_{f}^2 = m_{1}(u_{i} - u_{f})(u_{i} + u_{f}).

Then I find v_{f} using only u_{i}, and u_{f}:
v_{f} =\displaystyle{\frac{-(u_{i}+u_{f})-(u_{i}+u_{f})}{-2}}


Now, I have to substitute what I just found for v_{f} into the conservation of momentum formula, and solve for u_{f}. ...But I guess I'm having difficulty singling out the u_{f}.

This is what I've got:

m_{1}u_{1} = m_{1}u_{f} + m_{2}(\displaystyle{\frac{-(u_{i}+u_{f})-(u_{i}+u_{f})}{-2}})
...And here, I think I messed my algebra up, but when I simplify that, I get:
m_{1}u_{1} = m_{2}(u_{f} + u_{i}) + m_{1}u_{f}

Is that right? ...If not/if so, how do I get the u_{f} on just one side?

 

[Doc Al]

So-- I find m_{2}v_{f} using the law of conservation of momentum:
m_{2}v_{f} =m_{1}u_{i}-m_{1}u_{f}

OK.

And I find m_{2}v_{f}^2 using the law of conservation of kinetic energy:
m_{2}v_{f}^2 = m_{1}(u_{i} - u_{f})(u_{i} + u_{f}).

OK.

Then I find v_{f} using only u_{i}, and u_{f}:
v_{f} =\displaystyle{\frac{-(u_{i}+u_{f})-(u_{i}+u_{f})}{-2}}

I have no idea what you are doing here. This expression simplifies to: v_{f} = u_{i}+u_{f}, which is incorrect.

You can rewrite the momentum equation to get
v_{f} = \frac{m_1}{m_2} (u_{i} - u_{f})
Perhaps this is what you meant? Now just plug that into the KE equation to eliminate v_f and simplify.

...And here, I think I messed my algebra up, but when I simplify that, I get:
m_{1}u_{1} = m_{2}(u_{f} + u_{i}) + m_{1}u_{f}

Is that right? ...If not/if so, how do I get the u_{f} on just one side?

Amazingly, it is right. (It's equivalent to what you get when you do the "plugging in" that I suggest above.) So I suspect you made a typo earlier on. To simplify this expression, just multiply it out and move all terms containing u_f to one side and all other terms to the other side.

 

[verd]

woo, okay. thanks. I got it. ...Really, much appreciated.

Thanks again... (Eh, I'm having a really hard time in this class- if you couldn't tell. AND I made the mistake of taking it as a 7-week course. ...I obviously didn't know what I was in for. Really, thanks again.)