I was collecting data for a simple physics lab today when I stumbled upon a question I couldn't answer.(adsbygoogle = window.adsbygoogle || []).push({});

Very basically, the lab consisted of measuring the time it takes a ball to drop a variety of distances, between 0 and 100cm. By plotting [tex]\frac{y}{t}[/tex] vs. [tex]t[/tex] (where y is height and t is time) and running a LSQ fit, you can find the value of g. I want to take data at 20 different heights between 0cm and 100cm.

[tex]

y = V_0t + \frac{1}{2} gt^2

[/tex]

so [tex]g[/tex] is going to be proportional to [tex] u = \frac{y}{t^2} [/tex] where [tex]u[/tex] is [tex]\frac{1}{2}g[/tex].

Running error propagation:

[tex]

du^2 = (\frac{\partial u}{\partial y})^2 dy^2 + (\frac{\partial u}{\partial t})^2 dt^2

[/tex]

[tex]

(\frac{\partial u}{\partial y}) = \frac{1}{t^2}, and (\frac{\partial u}{\partial t}) = \frac{2y}{t}

[/tex]

So obviously of my 20 heights I should take more data down low and less data up high, because the longer the ball is falling, the less the error becomes, whereas the error in y is constant regardless of height.

Now my question is this: ideally, how should I space the 20 heights at which I will take data? What I'm basically asking is how to analytically solve for constant [tex]du[/tex] when the range and number of data points to be taken is given.

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# Error propagation and optimum

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