Numerical differentiation: Position → Velocity → Acceleration

Often in an experiment, some quantity is measured as a function of time. For instance, by videotaping a moving object it is possible to determine the position of the object in every frame of the videotape. Using a Doppler radar, it is possible to record the velocity of an object as a function of time. Using the accelerometer in a mobile phone, it is possible to record the acceleration as a function of time. If one of these quantities is measured, the others can be calculated. If the measured data is in tabular form, the integration or differentiation must be done numerically.

This page contains some programs differentiating numerical data. One component of the position vector (for instance $r_x(t)$) can be specified by pasting two columns of data into the red textbox. When the "calculate from table" button is pressed, the data is plotted on the right. Below the data and plot of $r_x(t)$, the velocity $v_x=\frac{dr_x}{dt}$ and the acceleration $a_x=\frac{d^2r_x}{dt^2}$ are tabulated and plotted. This process can be repeated for all three components of the position vector.

It is also possible to fill the red textbox with data by typing a formula into the blue textbox and pressing the "Calculate from formula" button. This formula is used to fill the table with 1000 equally spaced values of $r_x(t)$ equally spaced between $t_1$ and $t_2$.