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

## 2/20-22/18 Tues Thurs Phy2400 Lab[1]

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  do we really need this info? 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 ±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[2][3]

 t (s) x (m) v a (m/s/s) * by hand -0.5 0 * t* v* 0 0.94 1.6 -1.99 * -0.25 1.88 0.5 1.6 * 0.25 1.32 mot. dec. 1.6 2.113 v_ave 2 -0.7875 * 2 1.798 2.4 1.483 average 1.7 -1.4 ± 0.1 0.6 m/s m/s/s

#### by hand

${\displaystyle v_{n}={\frac {x_{n+1}-x_{n+1}}{t_{n+1}-t_{n-1}}}}$

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

${\displaystyle v_{f}={\frac {x_{n+1}-x_{n}}{t_{n+1}-t_{n}}}={\frac {x_{n+1}-x_{n}}{\Delta t}}=v_{f}\left(t={\tfrac {t_{n+1}+t_{n}}{2}}=t_{n+{\tfrac {1}{2}}}\right)}$

The latter notation identifies when we have estimated vfinal

${\displaystyle v_{i}={\frac {x_{n}-x_{n-1}}{t_{n}-t_{n-1}}}={\frac {x_{n}-x_{n-1}}{\Delta t}}=v_{f}\left(t_{n-{\tfrac {1}{2}}}\right)}$

1. This lab was performed and written in two parts. The Tuesday group wrote individual reports and collected the data. The second group wrote the following summary after reading the individual reports and holding a group discussion with Guy vandegrift.
2. By hand: 120 beats per minute. 0-94-160cm. pingpong roll
3. vernier: 2.113m/s at 1.6 s, and 1.483 m/s at 2.4 s (read off the graph on the hand-held device)