In which we estimate the exit velocity and flight time from our Mark 1 prototype, and use that to estimate the possible range at 45 deg launch

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In which we estimate the exit velocity and flight time from our Mark 1 prototype, and use that to estimate the possible range at 45 deg launch

library(ggplot2)

shot.dist <- 67 #meters
angle.deg <- seq(from=5, to=20) #Unknown / estimated
angle.rad <- angle.deg/180 * pi
G <- 9.8 #mpss accel from gravity

# Given angle theta, we have horizontal velocity of cos(theta)*v_0
# So, cos(a)*v_0 * t = 67 -> v_0 = 67/[cos(a)t]
# And sin(a)*v_0 - gt/2 = 0 at the peak, so 
# 67 s(a)/[c(a)] = gt^2/2
# 134* tan(a)/[g] = t^2
t <- sqrt( 2* shot.dist * tan(angle.rad)/G )

# Revisiting 
# sin(a)*v_0 - gt/2 = 0 at the peak, so 
v_0 <- G*t / 2 / sin(angle.rad)


ggplot()+
geom_point(
    aes(
        x=angle.deg,
        y=t
    )
)+geom_line(
    aes(
        x=angle.deg,
        y=t
    )
) +labs(
    title="Estimated flight time as a function of launch angle with known distance traveled"
)

ggplot()+
geom_point(
    aes(
        x=angle.deg,
        y=v_0
    )
) +
geom_line(
    aes(
        x=angle.deg,
        y=v_0
    )
)+labs(
    title="Estimated exit velocity as a function of launch angle with known distance traveled"
)

# Hypothetical travel at 45deg
# sin(45)*v_0 - gt/2 = 0
t.45<-sin(pi/4)*v_0/G*2
dist.45 = cos(pi/4)*v_0*t.45
summary(dist.45)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  104.2   124.8   158.8   186.7   223.4   385.8

[Jodavaho.io]

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