Physics and Astronomy Labs/Air drag coefficient on a rolling ping-pong ball

==2/20-22/18 Tues Thurs Phy2400 Lab == The purpose is to measure the air drag coefficient of rolling ping-pong ball that acquired it's speed by rolling it down a plastic ramp. It was necessary to keep the ball on the ramp by extending it on the flat surface Two simple methods were used. The simplest involved measuring the position at three consecutive times and using finite difference calculus to estimate the acceleration. The other method involved a Vernier motion detector that obtains values at three consecutive times produced graphs of acceleration and velocity.

It was necessary to keep the ball on the ramp by extending it on the flat surface. The ramp was plastic and roughly one inch in diameter. When the ball hit a joint between two segments, it bounced with amplitude of less than a millimeter. We don't think our error in position was much more than &plusmn;1 cm. We attempted to estimate the percent uncertainty in acceleration using this estimate, but something went wrong with the spreadsheet used to randomly change the inputs.

 Results 

by hand
$$v_n=\frac{x_{n+1}-x_{n+1}}{t_{n+1}-t_{n-1}}$$

$$a_n = \frac{x_{n+1}-2x_{n}+x_{n+t}}{{\Delta t}^2}$$ where $$\Delta t = t_{n+1}-t_n=t_n-t_{n-1}$$

$$v_f=\frac{x_{n+1}-x_n}{t_{n+1}-t_n}=\frac{x_{n+1}-x_n}{\Delta t}=v_f\left(t=\tfrac{t_{n+1}+t_n}{2}=t_{n+\tfrac 1 2}\right)$$

The latter notation identifies when we have estimated vfinal

$$v_i=\frac{x_{n}-x_{n-1}}{t_{n}-t_{n-1}}= \frac{x_{n}-x_{n-1}}{\Delta t}=v_f\left(t_{n-\tfrac 1 2}\right)$$