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E-bike Physics

Image of German-designed e-bike via Wikimedia

There’s a new class of electric bicycles prompting oodles of attention, collecting millions of dollars in funding and providing little more than dust-collecting, spider web-encrusted racks in bike shops.

Within the last five years, Panasonic and MIT unveiled e-bike prototypes, which boast batteries that can re-charge themselves through a process called regenerative braking. A regenerative brake reduces a vehicle’s speed, thus expending energy, but in the process it captures some of that lost energy and stores it in the battery for future use. Contrary to what they claimed in 2008 and 2011, neither Panasonic nor MIT has released their design to markets.

There are many possible explanations, but one that stands out above the rest is simply that the design does not work, or at least does not work well-enough to compete with e-bikes already available for purchase. This post explores some of the physics behind this design and whether it’s capable of acting as an efficient e-bike.

E-bikes make biking faster and easier through the assistance of an electric motor that runs on rechargeable batteries. Surly Bikes, a designer and importer of bicycles based in Bloomington, Minnesota sells e-bikes up to $4,500 that can motor you along for 35 miles at maximum speeds of 20 mph before losing charge. Therein lies the problem – batteries inevitably drain out and e-bikers must bike the old-fashioned way to a power source so they can recharge their battery.

But, what if there was an e-bike that recharged your batteries as you biked? That’s the concept behind Panasonic’s Lithium Vivi RX-10S and MIT’s Copenhagen Wheel. Using regenerative braking, an e-bike can recharge its batteries when braking so an e-biker will allegedly never have to worry about recharging their batteries ever again.

Let’s take a closer look at our novel e-bike and biker. Ultimately, we want to determine whether regenerative braking can feed the batteries enough juice to accelerate the system (e-bike + biker) in traffic and up hill.

Say the e-bike, of weight 15 pounds (7 kg), and biker, of weight 150 pounds (68 kg), has just entered a traffic-heavy area and needs to accelerate. She’s traveling on flat ground at 10 mph (4.5 m/s) and wants to boost up to 20 mph (9 m/s) over a period of 10 seconds. Her final energy is modeled as:

Where we want to calculate E(motor).

Her initial and final energies are equal to her initial and final kinetic energies. The main component acting against her is aerodynamic drag and accounts for 70 to 90 percent of resistance that bike riders feel when pedaling on flat pavement. The work done on the system by drag, which is equivalent to E(drag), is modeled:

We went with a spherical-cow-inspired estimation of a bike rider as a small circle of radius 7.5 centimeters atop a rectangle of height 1.5 meters and width 0.4 meters. These numbers correspond to average head breadth, shoulder breadth and height of a woman on a bicycle. Plug the numbers into the final equation for E(motor) and we get:

Regenerative braking is common in hybrid cars where it can capture up to half of the energy dissipated while braking. Suppose an e-bike can capture a similar amount. The energy dissipated while breaking from 20 mph to 0 is equal to the initial kinetic energy of the system, which is 3037 Joules. That means the batteries could potentially store 1519 Joules – less than half the energy required to boost the e-biker along a flat road on a windless day with an engineer’s fantasy motor that runs at 100 percent efficiency.

There are also other factors to consider like the weights of the rider, the bike, the quality of the batteries and whether the bike is traveling up an incline. If hills are involved, the biker will require still more energy to travel at the same speed as they would if on flat ground.

Of course, if the batteries are not entirely dead, this could add a nice amount of extra charge. MIT’s design, the Copenhagen Wheel, recently received $2.1 million to help get their design to market and will be announcing new hardware revisions at the end of November, according to MIT News Office. Seems that the design still needs some work.

For other calculations concerning e-bikes with regenerative braking and other self-sustaining e-bike models visit:


  1. Surely this isn't news to anyone who wasn't expecting magic. The only issue here is the obvious fact that regenerative braking cannot recapture more than a fraction of the kinetic and potential energy, so all of the work against friction (plus a significant part of the energy creation) must be done either by the rider or by the initial charge on the battery. It is almost misleading to introduce a complicated theoretical calculation of something which can be easily found by experiment - the frictional energy requirement is just the work you need to do to maintain a constant velocity on a flat surface, and the lifetime of the unassisted battery will never be greater than it would be if used to run the bike at constant speed until it runs out. But for a ride that involves frequent stopping and starting, regenerative braking may save a significant fraction of the extra work involved in that aspect - and I believe that it has been found to significantly extend the battery life of e-assist cargo trikes in a downtown delivery setting.

  2. Nihil novi sub colem!
    I ride Stromer e-bike with 2 regenerative modes. At times I pedal with only regen mode (makes pedaling the 64lb bike quite an effort) The amount of returned energy is rather small, and I can't imagine recharging the whole battery while riding even for miles.

  3. I've wanted regenerative braking for some time on an e-bike. I'll go down a hill at 40km/h and using some brakes to not go faster than that. I'll then up another at 10km/h. Ideally, I'd go down the first hill at 30km/h, filling up the battery, and then use a little of that to go up the next hill. Yes, the process is inefficient, losing 50% of energy into the generator... but that's still 50% more than what goes into my brakes today.

    The downside, of course, is at least 10kg of extra equipment to lug up the hills!

  4. Yeah, regenerative breaking is very terrain and condition dependent. Best case scenario you can recover the energy up to 60 % of what you spent uphill. BUT this is really in ideal conditions, very steep downhill and hard breaking. They may have invented this for San Francisco. It would barely work in the Netherlands nor in Finland (well actually most of Europe).

  5. When Copenhagen wheel first came out there seemed to be a silly 'green' focus on absolute silly things, such as 'regenerative breaking' and 'environmental sensors' (Copenhagen wheel originally included CO2 and NOX sensors so you could share info on how 'clean' your route is.

    It looks like there were lots of idealists running the show, and few people who actually understood business, which has held them back and stagnated them for years. Even now, the specs of Copenhagen wheel, supposidly in production, are a secret. They won't release the energy capacity of the batter. The most recent images showing any of the internals are from 2009.

    Also a new compeditor has emerged, Flykly, and suddenly, in resposne to Flykly, which as been behind closed doors for over 4 years, is suddenly ready to roll-out at the same time as Flykly...

    I think we will have to wait for reviews of both to judge them. But one thing is already well known in the ebike community: You can capture about 30% of downhill gravitational energy. With a good motor, you can end up getting about 20% power back. Basically if you go down a 500m hill slow enough, can you get enough energy to be driven up a 100m hill.

    Also none of the bikes that are coming out will actually implement rego like this. At most they will simply have a capacitor that charges from regen and supplements the battery. Recharging the batteries slightly every time you go down a hill will dramatically reduces their lifetime. No one is going to want a 10% to 20% increase in range from regen if it means a shorter battery lifetime. The battieres are already only about 1000 cycles. If you use regen you could be looking at a new $500 battery back every year.

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