# Calculating velocity and acceleration from (timer reading, position) data using CAPM

I’m collecting some calculations I find useful for evaluating student data and for more advanced students to process their own data. CAPM says that we think acceleration is basically constant, gives us a mathematical model for how position and velocity depend on time, and allows us to make predictions. If we believe that CAPM holds, then we only need three points to calculate velocity and acceleration, one data point at the spacetime event in question and one on either side of the event in time.

## Assumptions

1. Acceleration is constant. Thus $x(t)=\frac12 At^2+Bt+C$ and $v(t)=At+B$.
2. The data point in question is $(t_0,x_0)$, and the points on either side are $(t_0-dt_-, x_0-dx_-)$ and $(t_0+dt_+, x_0+dx_+)$.

## General case

• $v(t_0)=\frac{\frac{dx_+}{dt_+}\cdot dt_- + \frac{dx_-}{dt_-}\cdot dt_+}{dt_-+dt_+}$
• $a(t_0)=\frac{\frac{dx_+}{dt_+}-\frac{dx_-}{dt_-}}{\frac12(dt_-+dt_+)}$

## Breadcrumb method case ($dt_-=dt_+=dt$)

• $v(t_0)=\frac{dx_-+dx_+}{2dt}$
• $a(t_0)=\frac{dx_+-dx_-}{dt^2}$

## Split time method case ($dx_-=dx_+=dx$)

• $v(t_0)=\frac{dx(dt_-^2+dt_+^2)}{dt_-dt_+(dt_-+dt_+)}$
• $a(t_0)=\frac{2dx(dt_--dt_+)}{dt_- dt_+(dt_-+dt_+)}$ (Note: This needs some reinterpretation.)

## Summary for use

With N data points, one can easily calculate N-2 velocities and accelerations at the same timer readings. Of course, one can calculate N-1 velocities at intermediate timer readings, but this can make it difficult for students to make spreadsheets for velocity. Using the method above, one can calculate both instantaneous velocity and instantaneous acceleration at the mesh of timer readings by focusing on three data points at a time. This makes it easier to test the validity of the Constant Acceleration Particle Model (CAPM).

I like to throw the formula into a spreadsheet that I use when looking at student data.  When they whiteboard in class, I type in their data to see how consistent it is, helping me to assess where students are going wrong and whether their graphs make any sense.