Dot Physics

Energy in a hidden battery

Back to the discussion about hiding an electric motor in a pro racing cycle. Before, I looked at a video of Fabian Cancellara to see how his speed and acceleration compare to other bikers. The claim on the internet is that he pulls away so fast that he must have a motor hidden in his bike.

Just to be completely clear, I don’t think he is cheating. Then why bother? If you ask that then this must be the first time you reading this blog. I welcome you. No, but really, this is what scientists do. How hard would it be to cheat?

From my analysis, it seems that a person could ride like he does without a motor. But, the question is: how much power for how long could he get if he DID hide a motor? I am going to assume that you could put some batteries in just two parts of the bike. Here is a typical bike with some estimated measurements.


The other post is where the motor would go – so no batteries there. Now, how big is the inner diameter of these tubes? I am going to estimate the outer diameter at about 4 cm. I guess a 3.5 cm inner diameter is not unreasonable. Right? So, what volume of space for batteries are we dealing with here?


Next question – what kind of battery would I put in there? Here is Wikipedia’s page on energy density. In the chart, there is a Lithium ion nanowire battery with an energy density of 2.54 MJ/kg. I don’t know anything about this kind of battery. I would use this, but I don’t even know the mass density (or the volume energy density). The next highest on the list is a Flouride ion battery with an energy density of 1.7 MJ/kg or 2.8 MJ/L. Just for comparison, Wikipedia lists the lithium ion battery with an energy density of 0.7 MJ/kg and 0.9 MJ/L.

If I used the fluoride ion battery, how much energy would I have stored?


That seems like a lot of energy. But now, what if I want to run a 500 watt motor? How long would that last?


Well, what about the mass of this battery? Using the mass energy density and a total energy of 2.8 MJ:


1.6 kg seems really too low for the mass. That is a weight of just 3.5 pounds. I guess you would also have the weight of the motor, but that shouldn’t be too much. So, maybe if you are James Bond and have access to this kind of stuff, this is possible. (I still don’t think he cheated).

What if you do not have access to these Fluoride ion batteries? What if you just use lithium ion? I would only have 0.9 MJ of energy which would last 30 minutes at 500 watts (oh, assuming 100% efficiency – which is obviously not going to happen). Also, this battery would have a mass of 1.2 kg. Not too bad.


  1. #1 jg
    June 11, 2010

    I feel like an extra 3.5 lbs isn’t worth the trouble. How about the rest of the race, where you’re carrying the dead weight? How long would the race need to be to make the motor not worth the benefit? Sorry, no time to crunch the numbers now. :)

  2. #2 Jared W.
    June 11, 2010

    Agreed jg, 3.5 lbs is A LOT of extra baggage when talking about elite cycling. Not worth the trouble. Additionally, any bike weighing that much during inspection before the race would be quickly noticed and talked about.

    Love the analysis Rhett. Fun to get my learn on. Thanks.

  3. #3 Thomas
    June 11, 2010

    You also have to worry about the noise from a 500 W motor.

    It would be kind of cool if it were true, though. There is something special about those blatant cheaters. Scoring a goal with your hand in football, installing an extra button in your foil, or best of all, take the subway during a marathon.

  4. #4 Anonymous Coward
    June 11, 2010

    In any discussion of whether or not 3.5 lbs is “worth the trouble” or not, you need to figure out what numbers you’re comparing.

    Typical mechanical power outputs for cyclists during the hill-climbing stages of the Tour de France are on the order of 400 to 500 Watts. Peak outputs at various times are probably close to 1 kW.

    So, by Rhett’s numbers, if a 500 W battery could add 100% of the bicyclist’s average power for 90 minutes, while adding less that 3% of the bicyclist+bicycle weight, I can’t see how that would not be worth it, especially in a shorter stage.

    Of course the analysis neglects motor weight and that jazz, but instead of “not too bad” or “not worth it”, the numbers in the analysis would suggest “incredible advantage”.

  5. #5 Martin
    June 12, 2010

    So, does this mean you’re going to declare the DDWFTTW cart to be a hoax, because there is enough room in the cart to hide batteries?

    (Never mind the fact that the cart’s build is well documented and anyone is invited to inspect it).

  6. #6 John
    June 13, 2010

    500w for 93 minutes would be more than plenty for a 20 mile round trip bike commute (if you’re not too much of a leadfoot), including a generous buffer for battery degradation or side trips!

    Jared’s right! The weight would be a dead giveaway. They’d get nailed in inspection for sure; these bikes are already trimmed down to where they’ll only handle a handful of races at most. Add the weight of a motor and batteries and frame reinforcement to handle the motor’s load, and you have enough of a deviance to warrant a full disassembly before the race. But if it sneaks through and the rider wins… There’s an inspection after the race, too. The leader’s bike doesn’t exactly get by with a quick look-over.

    Cyclists can do things like this; it’s part of the strategy. If the legs alone can keep one in the pack until breakaway time, a cyclist can use his/her arms deadlift-style against the bars for a burst of acceleration well into a race. But it’s a gamble – hang back a little, save that energy for your hat trick, and hope the leaders don’t peel off over the horizon early on.

  7. #7 travc
    June 13, 2010

    Interesting back-of-the-envelope calc. However, I think that the load capacity (or whatever you call the rate power can be delivered) is a pretty decisive factor.

    When you’re looking for high rates of power at very low weights, I think super/ultra-capacitors are a favored approach (if money isn’t a tight constraint.)

    For a “cheat” bike, assist just on the acceleration with very low extra weight seems like a more plausible way to go. The motor weight seems like it would be the biggest challenge.

  8. #8 Andy
    June 29, 2010

    One thing to note when asking if it’s “worth the weight” is that there is a minimum weight limit set. Some manufacturers have managed to build frames that are lighter than this limit, and then they typically add lead weight to bring it to spec. If, for example, they managed to build it 3.5 lbs below the minimum, they would have to add 3.5 lbs of weight. At that point there is no weight penalty for adding the motor, since that weight would be added anyway, as ballast.