# Unit conversion

If I had to condense advice for life/wellbeing/career into a soundbite, I’d go with “be curious”.

I’ve been comparing cycle routes on CycleStreets, Google Maps, and Strava. CycleStreets has a energy cost estimate of the journey, with a little ? link next to it.

They outline their estimate of calorie cost of a journey, including their formula.

This is a brilliant example of the mess of units we use. The formula was written for meters/second, I think of road speed in miles per hour, a lot of the world thinks of kilometres per hour. I think of human energy in kcal, the formula wants to use watts.

All my route finders give a total ascent/descent stat, but if I want my distances in miles they give it in feet, when I think of ascent in meters.

I have R-bloggers in my News Reader. A while ago the authors of the package announced units on R-bloggers.

I had my formula in science units, my head in a mess of units, but I remembered that I spotted units had been released.

## Back to energy spent cycling

A first stab at making that formula as a function:

power = function(v,
g=9.81,
m=90,
k_1=0.0053,
k_2=0.26,
s=0){
g * m * v * (k_1 + s) + k_2 * v^3
}

power(5.4)
## [1] 66.20924

I simplified it a bit by saying v_a = v_g. And we’ve replicated their example.

If I do the sums, then apply units I can look at a speed-power curve:

tibble(speed = seq(0, 11, length.out= 100)) %>%
mutate(power = power(speed)) %>%
mutate(speed = set_units(speed, "m/s")) %>%
mutate(power = set_units(power, "watts")) %>%
mutate(speed = set_units(speed, "miles/hour")) %>% #SI to thinking in road speed
mutate(power = set_units(power, "kcal/hour")) %>% #SI to human-level energy units
ggplot(aes(x=speed, y=power)) + geom_line()

This is after I tried starting with speed in miles/h and converting to m/s. It didn’t like that because the units are incompatible.

This is great! Sensible errors are better than producing nonsensical output, so let’s rewrite the function including a statement that most of these parameters have dimension.

power = function(v,
g=9.81,
m=90,
k_1=0.0053,
k_2=0.26,
s=0){
units(v) <- "m/s"
units(g) <- "m/s^2"
units(m) <- "kg"
units(k_1) <- 1
units(k_2) <- "kg/m"
units(s) <- 1
set_units(g * m * v * (k_1 + s) + k_2 * v^3, "watts")
} 

Then what I wanted for the above example is:

tibble(speed = seq(0, 25, length.out = 100)) %>% #I think this is clearer as 2 lines
mutate(speed = set_units(speed, "miles/hour")) %>%
mutate(power = power(speed)) %>%
mutate(power = set_units(power, "kcal/hour")) %>%
ggplot(aes(x=speed, y=power)) + geom_line()

## Other graphs

I read Sustainable Energy Without the Hot Air, which suggested in general that moving slower is more energy efficient for distance. 1

I have:

• speed $$\frac{m}{s}$$
• power $$\frac{j}{s}$$

I want:

• energy per distance $$\frac{j}{m} = \frac{j}{s} \div \frac{m}{s} = \frac{\text{power}}{\text{speed}}$$

So the dimensional analysis helped me work out what sums I wanted to do.

tibble(speed = seq(0, 25, length.out = 1000)) %>% #I think this is clearer as 2 lines
mutate(speed = set_units(speed, "miles/hour")) %>%
mutate(power = power(speed)) %>%
mutate(energy_per_dist = power/speed) %>%
mutate(energy_per_dist = set_units(energy_per_dist, "kcal/mile")) %>%
ggplot(aes(x=speed, y=energy_per_dist)) + geom_line() +
ggforce::scale_y_unit(breaks = c(0,5,10,15)) + expand_limits(y=c(0,15))

Or using “fat is 9kcal/gram”:

tibble(speed = seq(0, 20, length.out = 1000)) %>%
mutate(speed = set_units(speed, "miles/hour")) %>%
mutate(power = power(speed)) %>%
mutate(energy_per_dist = power/speed) %>%
mutate(energy_per_dist = set_units(energy_per_dist, "kcal/mile")) %>%
mutate(fat_burned = energy_per_dist / as_units(9, "kcal/g")) %>%
mutate(fat_burned = set_units(fat_burned, "g/Mm")) %>%
ggplot(aes(x=speed, y=fat_burned)) + geom_line() +
expand_limits(y=0)

“Mm” being “mega-meters”, or 1000 km. Somewhere in the region of 40 office commutes = 1Mm, for me. I suppose this reinforces the truism that one does cardio to improve their cardiovascular health, not for the energy deficit!

## Conclusion

• The {units} package can simplify a bunch of analysis, directly by handling conversion, and indirectly by pointing out that you’re trying to add meters and seconds.
• Cycling can be part of my weight loss, but not the whole story.

1. That formula sure looks like it’d apply to a motor vehicle, with some different k_1 and k_2.↩︎