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Phy 231
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
derivative- cons of momentum
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This is in regards to the portion of the Conservation of Momentum lab that asks us to find an expression for mass ratio and then take its derivative.
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I feel fairly confident about my expression:
m1/m2 = u2/(v1-u1)
Then your instructions say to treat u2 and u1 as constants and take the derivative with respect to v1 as a variable. First, I find myself wondering why this is. What about the situation makes v1 any more logical as a variable than the others?
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v1 is the one quantity you have control over. You can vary v1 by varying the slope of the ramp or the distance the ball moves along the ramp.
The derivative with respect to v1 will give you some indication of how an uncertainty in your measurement of v1 would be related to the uncertainty in the mass ratio.
Of course uncertainties in u1 and u2 also affect the uncertainty of your final result, but analysis of the effects of all the uncertainties would require some multivariable calculus, which is a step beyond the scope of this course. So we confine our attention to the effect of the uncertainty in v1.
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Then, I wonder how even to set up the derivative.
My assumption is that, since we want the derivative with respect to v, I would call the original function f(v) and the derivative f'(v). So here's what I did:
f(v)=u2/(v1-u1)
f(v)=u2(v1-u1)^-1
f'(v)=-u2(v1-u1)^-2
or
f'(v)=-u2/(v1-u1)^2
Plugging in the numerical values I found, this gives me:
f'(v)=-101.6cm/s / (64.6cm/s-54.3cm/s)^2
f'(v)=-101.6cm/s / (10.3cm/s)^2
f'(v)=-101.6cm/s / (106.1cm^2/s^2)
f'(v)=.95/(cm/s)
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The derivative is the rate of change of the unitless ratio m2 / m1 with respect to v1, so it is the limiting value of the average rate
ave rate of change of ratio with respect to v1 = (change in ratio) / (change in v1).
The change in the ratio is unitless, the change in v1 is in cm/s, so the rate has units of (cm/s)^-1.
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These units don't make any sense to me. I would expect to find final units in cm/s^2, as taking the derivative of a velocity function should lead me toward an acceleration function. But then, as a mass ratio, this didn't even look or seem like a velocity function in the first place.
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The function u2 / (v1 - u1) is unitless. Its derivative with respect to v1 has reciprocal units of v1.
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So maybe my problems are unit-related, or maybe something bigger. I can't grasp what the significance of this .95 might be, if it's even accurate.
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Go back to the idea
rate = (change in ratio m2/m1) / (change in v1).
If the rate is .95 (cm/s)^-1 then, for example, then what would be the implications of a +- .5 cm/s uncertainty in v1?
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