altura<-c(2.25,2.00,1.80,1.70,1.60,1.50,1.25)
tiempo<-c(0.729,0.690,0.623,0.607,0.586,0.553,0.503)
h<-seq(1.0,2.50,0.1)
t<-sqrt(h*2/9.8)

t2=tiempo*tiempo

plot(t2,altura,xlim=c(0.4,0.90),ylim=c(0.8,3.0),main="Altura(cm) vs Tiempo(s)",xlab = "t(seg)",ylab="altura(cm)")
lines(t,h,col="black")
abline(-0.8785,4.2723,col="red" )
abline(v=0.7,col="yellow")
abline(h=2.0,col="blue")
abline(v=0.5,col="blue")

# Linealizado altura vs \(t^2\)

altura<-c(2.25,2.00,1.80,1.70,1.60,1.50,1.25)
tiempo<-c(0.729,0.690,0.623,0.607,0.586,0.553,0.503)
h<-seq(1.0,2.50,0.1)
t<-sqrt(h*2/9.8)

t2=tiempo*tiempo

plot(t2,altura,main="Altura(cm) vs Tiempo(s)")
abline(0.4698,3.3319)

modelo<-lm(altura~t2)
summary(modelo)
## 
## Call:
## lm(formula = altura ~ t2)
## 
## Residuals:
##          1          2          3          4          5          6          7 
##  0.0088906 -0.0526873  0.0468317  0.0138371 -0.0008638  0.0271105 -0.0431189 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.43169    0.06582   6.559  0.00124 ** 
## t2           3.40475    0.16828  20.233 5.45e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03958 on 5 degrees of freedom
## Multiple R-squared:  0.9879, Adjusted R-squared:  0.9855 
## F-statistic: 409.4 on 1 and 5 DF,  p-value: 5.454e-06
# $\sigma$
# $\alpha^{3x}$
# $\int{\frac{3a}{1-x}}$
# $\sqrt{9x^2}$