Uncertainty in Acceleration given the uncertainty in position

I had a motion detector record the position of a dynamics cart and automatically plot Position vs Time and Velocity vs Time plots in Logger Pro on the computer. If the instrument uncertainty in position $x$ was given by $±0.05$ (or more generally any value $±e$) what will be the propagated instrument uncertainty on the second derivative of position, $\ddot{x}$ -- i.e. acceleration?

Thanks in advance

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Hi and welcome to this site. In general, we don't just answer a question like yours, because we like to see you put some work in there. Tell us what you know about uncertainty propagation and also explain why and where you are stuck. You can edit your question to add this information. – Lagerbaer Mar 25 '13 at 14:54
@Lagerbaer Hello and thanks of your answer. The only thing I know about uncertainty propagation is how to find the uncertainty in the gradient of a line or how to find uncertainties of multiplied and summed quantities. That's because I am still at school, and I am doing this for a lab I have to complete. As said, the sensor of known uncertainty measured the position of the cart as it slided down a ramp and a plot of Velocity vs Time was automatically generated by the program on the computer. Then I found the slope at the linear portion of this plot, which is the acceleration of the cart. – Ryuky Mar 25 '13 at 15:01

1 Answer

"The only thing I know about uncertainty propagation is how to find the uncertainty in the gradient of a line or how to find uncertainties of multiplied and summed quantities."

Well, there you go: The velocity is calculated from the positions by taking the difference of two positions and dividing by the time step $\Delta t$ of your measurement setup: $$v(t) \approx \frac{x(t + \Delta t) - x(t)}{\Delta t}$$ Since you know how to find the uncertainty for summing quantities, you can now calculate the uncertainty in velocity rather than position.

Next, acceleration is found from the gradient of the velocity line, and you say you know how to find that uncertainty, so you should have all the ingredients.

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