The rate of change of p with respect to time, dp/dt is a
vector whose components are the derivatives of px and py. To
avoid a lot of calculus let's make up a table symmetrical
about t=2.0 and calculate the values of px and py for each time
in the table. Then from the finite differences between px and py,
calculate the rate of change of px and py, we call them px'
and py'. Then from the finite differences between px' and
py', calculate the rate of change of px' and py', we
call them px'' and py''.
t | 1.999 | 2 | 2.001 | ||
5t^{2}-6t | 7.986005 | 8 | 8.014005 | ||
cos(5t^{2}-6t) | -0.1316402 | -0.1455 | -0.1593413 | ||
sin(5t^{2}-6t) | 0.99129757 | 0.98935825 | 0.98722356 | ||
2t+10 | 13.998 | 14 | 14.002 | ||
px | -1.8426991 | -2.0370005 | -2.2310965 | ||
py | 13.8761833 | 13.8510155 | 13.8231043 | ||
px' | -194.30139 | -194.09605 | |||
py' | -25.167888 | -27.911159 | |||
px'' | 205.346831 | ||||
py'' | -2743.2711 |
I did this on a spreadsheet so I could try different delta t values. Decreasing delta t by a factor of 1000 only changed force in the fifth significant digit.
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