## Introduction

While biking in the city, I am frequently passed by electric and non-electric bicyclists. I glare at the offending passers-by for confirmation of tell-tale signs of an e-bike. Any large dark barnacles protruding from the frame will set me at ease. A motor hanging from the bottom bracket, a battery bolted to a downtube are usual suspects. Occasionally the e-cyclist soft pedaling at 20 miles per hour into a stiff headwind is sitting so upright I suspect their entire spinal column is fused together. Sometimes, I am forced to reckon with my own carefully constructed house of cards. A commuter passes me with no electric propulsion components in sight, no matter how hard I squint. Needless to say, my interest is piqued.

Of the many bikes I’ve watched whizzing by me, a few have caught my eye. Primarily the front-loading cargo bike.

Whether you want to carry 3 kids or a keg (both?), this platform has potential.

## Front-Loading Cargo Bikes

Despite the good looks of the aforementioned front loader, there is competition in the cargo-bike beauty pageant space. Objectively, the best looking front loading cargo bike is the Larry vs Harry (LvH) Bullit.

But are these things really *practical* for getting from A to B? Maybe not. A used car at the same price point, would have vastly more utility. Amongst many other flaws, the Bullit (and other cargo bicycles) have no stock weather protection for you or your cargo, they’re bound to be slow-going, and the majority of the United States transportation network is conveniently designed for car-goers. Of course, there is a niche space for which a cargo bike is probably more practical, but I only paint with broad brushes.

I still want one, and I want to know if it makes “sense” in a different sense. Primarily: How can I justify an electric bike over the purists approach?

## A Purists Approach

A purely human powered cargo bike is the romantic choice. It provides the right to be “that guy” who might casually bring up that he doesn’t own a microwave, or eat fast food, during an only tangentially related conversation. So whats wrong with the romantic choice? I’m pretty sure the answer is power.

Here is a simplified equation for how much power \(p\) it takes to propel a bicycle:

$$ p= \frac{v(F_A+F_S+F_R+F_I)}{\eta}$$

Where \(v\) is velocity, \(F_A\) is force required to overcome aerodynamic drag, \(F_S\) is force required to climb or descend a slope, \(F_R\) is force required to overcome rolling resistance, \(F_I\) is force required to accelerate, and \(\eta\) is drivetrain efficiency.

You can break down the equation further by filling in what each force requirement entails. For instance, the force required to overcome aerodynamic drag is:

$$ F_A = \frac{1}{2}\rho v_w^2 S C_D $$

Where \(\rho\) is air density, \(v_w\) is velocity through air, \(S\) is frontal area, \(C_D\) is drag coefficient.

If you break out all the force equations you get:

$$ p = \frac{v(\frac{1}{2}\rho v_w^2 S C_D+mgsin(arctan(s))+mgC_R+ma)}{\eta} $$

Mass shows up a lot, and so does velocity. I happen to know that slope \(s\) is a very important component too.

A problematic real-world scenario for the purist is biking up a steep road with a heavy load. So how much power does that take?

The bike weighs 25kg. I weigh about 80kg. I want to carry ~20kg of cargo. So the total mass to lug around is about 125kg. Air density is about 1.225 \(\frac{kg}{m^3}\). I’ll assume there is no headwind so \(v=v_w\). Frontal area is approximately 0.6 \(m^2\). Drag coefficient is 1. Coefficient of rolling resistance is .007. Acceleration is 0 \(\frac{m}{s^2}\). Road grade is 6% and drivetrain efficiency is 0.96. These assumptions give the following relationship between power and speed in red:

I threw some dashed lines of constant power on the plot at 200w and 400w. A reasonably healthy man can maintain 200w for an hour, but 400w for 2 minutes or less. This means that I could climb a 6% grade at about 5mph if I plan to make it up a long hill, assuming I have the gearing to maintain a reasonable cadence.

### Hills – A Purists Worst Nightmare

The available world to be cycled upon does not cap out at 6% grade. In fact, the first route I checked between two parks near to me shows a portion of road that hits a 9% grade:

At a 9% grade, 200 watts would have me going 3.5 mph, which is only practical with mountain-bike gearing. At 400 watts I’d travel a blistering 6.8 mph. This is “push your bike” territory. In any reasonably hilly place, needing well over 600w of power to ascend a hill at 10 mph is a buzzkill.

Here is a surface plot showing the power required to move me on a loaded cargo bike at 15mph with a variety of road slopes (grades) and headwinds:

With no slope, and no headwind, this front loader would take 200w of power to maintain 15mph on a smooth road. Adding any slope or headwind makes for extremely difficult conditions. The only conditions I could maintain 15 mph in are yellow (not much of the surface).

### Speed Relative to Traffic

A more hand-wavy issue with the purists approach is that speed of traffic is largely unrelated to road grade or headwind. If I ride a human-powered cargo bike up a hill or into a strong wind, I am relegated to single-digit miles per hour. Most car-goers aren’t interested in traveling at those speeds. I have a hunch that the bigger the difference in velocity between me and the cars, the riskier my travel becomes.

## Can an Electric Bike fill the gap?

The purist approach is out. I’m not interested in lugging cargo around the city under the power of my legs alone if it means I have to travel at a snails pace in mildly adverse conditions.

So what kind of power can ebikes put out? I know off-hand that many ebike motors are advertised based on their power rating. Someone might sell an ebike with a “500w motor” – since it generally isn’t clear if that means instantaneous maximum power before the motor explodes? Average power before it overheats in some standard conditions? Maximum output mechanical power? Maximum input electrical power? I’ll just assume that it means the motor is capable of putting out 500 watts of mechanical power for a reasonable duration.

This is great news – because 500w of power is enough to dramatically expand the operating envelope of me on a cargo bike at 15mph. If I look back at the power required graph, its like shifting the previous “yellow” limit, to “anything purple or brighter”.

A typical e-bike motor can provide a lot of assistance relative to our flesh and bone capabilities. That’s much appreciated when operating up hills, or into wind, or with a heavy bike. Above, I assumed that the 500w of the electric motor came with no added mass to the bike, which is a bad, but not particularly harmful assumption. In reality, a typical ebike might weigh 20lbs more than its non-electric counterpart, which is somewhat negligible considering the 500w of mechanical power it is capable of adding. In other words, ebike systems have very high power-density (output power / weight) when compared with a person.

### But what about the range, battery, controller, etc.?

For this post I am going to assume that the controller, battery, and other component selections are optimized for the mission. In my case, local destinations <30 miles round-trip, so no special considerations need to be made at a high level.

## Electric it is, so what’s on the market?

Here is what I could find on electric Bullits from two local shops:

And:

It is strange that they both arrived at the $7,200 lowest cost of entry number. This is pretty expensive for my level of interest, unless I can find some fun science-y things to do with the bike to justify it in my head. One shop has a Bullit frameset for sale for $2,400, and they occasionally come up on craigslist, so in a future post I hope to break down the cost and performance difference between a DIY ebike build and one of these production models.