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Acceleration proportional to time

Friday 1 October 2021

I have been having some interesting discussions with a student about how to create the example often used in maths of acceleration that is proportional to time. The two examples I came up with are a charge in between two parallel plates where the PD between the plates is increasing linearly with respect to time, and at a fixed distance from the centre of a white dwarf whose mass was increasing proportional to time (only holds for short distances). What we would like to achieve is work out an equation for the change of gravitational field such that a particle moving in pace will have acceleration proportional to time without having to change the field with respect to time. How would we have to change the universal law of gravity to make this happen. Another challenge is to create a slope with a curved surface such that a rolling ball will have acceleration proportional to time. Or how about a spiral track around a point mass? You can see in the animation that this gives an acceleration that inreases with time but it's not proportional.
I have had some success using an iterative model to plot the acceleration vs distance graph for this motion. This leads to the force distance graph which can be integrated to give the PE vs distance graph, in other words the potential well. I cut this out and put it in Algodoo but have some refining to do to make the ball stay on the surface.


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