The Data

The data is experimental. Students collected the data by attaching a thin piece of wax paper to a falling weight. As the weight falls, the paper goes through a spark generator which burns holes in the paper in \(\frac{1}{60}\)s intervals. Students measure the distances between holes with a ruler and calculate velocity: \(\vec{v} = \frac{\Delta \vec{y}}{\Delta t}\). Where \(\Delta y\) is distance between holes and \(\Delta t = \frac{1}{60}\)s.

Students then graph these data using EXCEL and find the least-squares regression line. The slope of the line will give them Earth’s gravitational acceleration \(\vec{g} = -980 \frac{cm}{s^2}\).

t = c(0:17)/60
e = rnorm(17,0,2.5) # add some random error so the measurements will be more realistic
v = 980*t
t
##  [1] 0.00000000 0.01666667 0.03333333 0.05000000 0.06666667 0.08333333
##  [7] 0.10000000 0.11666667 0.13333333 0.15000000 0.16666667 0.18333333
## [13] 0.20000000 0.21666667 0.23333333 0.25000000 0.26666667 0.28333333
v[2:18] = v[2:18]-e #Error mostly came from the length measurments which will cause velocity to vary randomly
# Removing these data represents a spark that failed to fire creating a gap in the data
v <- v[-14]
t <- t[-15]
df = data.frame(v,t)
suppressMessages(suppressWarnings(library(ggplot2)))
ggplot(df, aes(x = t, y=v))+
  geom_point(color = "orangered")+
  geom_smooth(method=lm)

fit <- lm(v ~ t)
summary(fit)
## 
## Call:
## lm(formula = v ~ t)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9781 -2.9272  0.1288  0.4893 16.7319 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.1763     2.3230  -0.076    0.941    
## t           987.4401    14.4166  68.493   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.111 on 15 degrees of freedom
## Multiple R-squared:  0.9968, Adjusted R-squared:  0.9966 
## F-statistic:  4691 on 1 and 15 DF,  p-value: < 2.2e-16

I realized that the spark generator had skipped a beat causing a jump in the velocity. Note that the overall effect on the result is about a \(\%e = \frac{1000-980}{980}\approx 2\%\).