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Formula for a Wheelie

Discussion in 'General Motorcycling Discussion' at netrider.net.au started by Shibboleth, Apr 26, 2013.

    In the CB400 thread, Positron007 speculated that the CB400 (which accelerates really quickly to 70kph) accelerates as fast as a 600cc sportsbike, at least up to the first gearchange.

    He speculated that the limiting factor (up to 70km/h) was not the power of the bikes (which are vastly different) but the fact that you’re trying to prevent the bikes from doing a wheelie for this whole time.

    So today, I thought I'd use my nerd powers for good instead of evil, and take a crack at explaining a wheelie formulaically. I couldn't really find anything explaining the physics behind a wheelie on Google, so I thought I'd have a crack at it myself.

    Disclaimer: May be wrong.
    TL;DR: Positron is right, 600cc bikes don't really accelerate faster (to 70kph) than the CB400.

    Assumption 1 - That both the CB400 and the 600cc sportsbike (which I'll call the CBR600RR from now on) are both at their 'liftoff point' for the whole drag-race (which will be up to 70km/h, after which the CB400 is not capable of wheelie-ing).

    Assumption 2 - The CB400 weighs 270kg fully laden with 76kg rider, and it's COG (with rider) lies 600mm and 500mm from the rear axle.

    Assumption 3 - The CBR600RR weighs 273kg fully laden with 76kg rider, and it's COG (with rider) lies 650mm and 550mm from the rear axle.

    Assumption 4 - Gravity is 9.81 m/s2

    Why is the liftoff point important? A bike is accelerating at its fastest when the front wheel is at the point of becoming weightless. If the bike is accelerating any faster than this, then the front wheel will continue to lift off the ground, until the rider is on his back. (See endless YouTube videos to see this effect in action).
    At liftoff point, both the liftoff and the gravitational turning moments are equal. What do I mean by this?


    To describe this as a formula:

    Moment (downwards) = Moment (upwards)​

    I don't think that this formula applies to drag-cars and other very-high performance vehicles. The principles do apply, but they also have some massive lifting forces created by the torquing action on their drivetrain. Bikes do not have torque-induced lift, especially at the low-power levels we're looking at here.

    The turning moment is the force multiplied by the perpendicular distance to the turning point, and can be described by the formula:

    Mt = F * D

    To show the turning moments visually:


    Horizontal Turning Moment


    Vertical Turning Moment

    Force is described by the formula

    F = m * a

    So, to combine all these equations


    Solving for acceleration provides:


    This is quite an interesting result, as the mass cancels out of both sides. Ergo, the mass of the bike has nothing to do with the bike’s ability to keep the front end off the ground. (Naturally, the bike’s mass will have a lot to do in limiting the amount of acceleration available to the bike.)

    So it all comes down to the D(h) against D(v) ratio. In retrospect, this makes sense, as it is well-known that an adequately powerful cruiser can out-accelerate a sportsbike, due to its low COG and long wheelbase (more forward COG).


    So comparing the CB400 against the CB600RR:



    So the CB400 will actually out-accelerate the CBR, though there is not much in it. These accelerations will mean that the bikes will do 0-70km/h in 1.7 seconds. NOTE: The numbers for the vertical and horizontal distances are pure guesswork. Any assistance to provide real numbers would be greatly appreciated.

    In Summary:
    • The maximum acceleration threshold for a bike is limited by its turning moment.
      • At any point below this threshold, acceleration will be limited by other factors, such as power, gearing, weight, etc.
    • The turning moment is a factor of the height and distance of the COG of the Bike (and rider) from the rear-axle.
    • Any two bikes which have sufficiently similar COG distances will have the same maximum acceleration threshold.
    • Like Like x 4
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  2. Mmmm, i think i may need a little longer to digest all that!
  3. I think if you line up a decent rider on a 600cc bike, they should look at their launching skills if they cant smoke a cb400 lol
    • Agree Agree x 2
  4. Get back in the pit, nerd, or it gets the hose again ...

    I kid, good work! I remember studying moments of complex structures at uni so I could follow this. Your theory is solid and results are backed up by centuries of physics hypothesis and knowledge. Kudos, sir :)
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  5. I'm with the Shibboleth on this.. Although it would be pretty even up to whatever speed the cb400 tops out at in first (61 km/h?), the 600 can probably keep going to 100 or more until it needs to change gear.

    When you think about it, it's the way it has to be really - the 600 has more power, but if you packed all of that into the same 60km/h range as the 400 it'd be quite a handful to ride. That's why the powerful litrebikes can get all the way to 150+ in 1st gear - if they were geared so that 1st only went to, say, 50km/h, they'd never have both wheels on the ground.
  6. Given the CB400 is a LAMS bike and a CBR600RR is not , you would expect, at best the CB to chase well up to a speed, but to never beat the CBR .
    If I had the CBR and a CB400 beat me it would mean time to sell or get some track days :)
  7. I would've preferred you guys settling this debate in the car park with a few races. I know positron has some experience with wheelies.. but fcuken nerds.. they prefer to race motorcycles on paper hahahha
    • Agree Agree x 4
  8. #8 Ljiljan, Apr 26, 2013
    Last edited: Apr 26, 2013
    You guys need a reality check.


    Now stop taking the piss.

    • Like Like x 1
  9. lol how long did that take???

    all good stuff but obviously some serious assumptions... both bikes have to have enough torque, and the ability to transmit it to the ground, to lift the front wheel at any point till 70kph....
    i'm just trying to think of how the launch might go dowm....this would require a LOT of clutch slipping to keep your revs in that sweet zone and i still reckon the CB400 would run out of puff... looking at the dynos from the two... a cbr600 has about 45ft.lbs, the CB has 25. I'm guessing you picked 70 cause of the first gear on the CB.. so taking into aco**** the CBR (which changes near on 100) will make a corrected 31ft.lbs..... so SLIGHTLY more (lots of assumptions here too)

    the thing is, Ive had a CBR600RR for a bit over 5 years... it does not lift the front wheel in first through engine power alone. You have to clutch it, or snap it, or bounce it or have a pillion. you can cheat a bit to get the front up by slipping the clutch, but as your speed picks up and catches up to your revs, and the engines power drops off or you hit the limiter... it's gonna come back down..... I can't see a CB400 having more on tap...

    I'm not saying you're WRONG... i just think 70 is probably optimistic, i suspect the CB400 might be faster to 40, before it reached the point it's power started dropping off and it couldn't keep the front up.... bearing in mind the calcs, and the best accelleration are with the front wheel still bairly touching... im sure if you clutched it up to balance point you could wheelie beyond that but thats not gonna be your best accelleration.

    plus the fact the CBR's ergos make the riders weight sit a bit further forward...
  10. Unless there is a spud on the 600, the CB might be quicker to 4 thats about it.

    600s are probably the easiest bikes to launch with the exception of big cruisers and v-rods etc.

    They have power, but are less wheelie prone than 1000s
  11. Actually this reminds me of a bloke in the same thread several years ago that swore his cb400 could keep up a zx6r all the way to 100.
  12. For some reason rogercordia popped into my head here lol
  13. First of all, I'm with blackadder; just go and line up somewhere and have a few test runs.. :)

    That said, I just had a play with the numbers and max torque for the cb400 (@9500 rpm) translates to about 820Nm at the wheel in first gear. The CBR600RR produces 930Nm at the wheel in first (@11200rpm).

    Even with the gearing differences, there's still about a 15% _at the wheel_ difference in the amount of torque that the 600RR is capable of laying down.

    _If_ that 15% is all excess that would result in a wheelie anyway, then maybe it's moot, but at the same time, maybe it's not.

    @trd2000 is right too, the 600RR will be nowhere near it's rev limit @ 60km/h, and will still be pulling hard, while the cb400 will be starting to get a bit asthmatic by about 50-55 probably..

    Meh.. go do a few test runs and see what happens.

    Bahaha... Pretty easy to disprove that one I would have thought..
  14. being fair though guys, without taking into account the engines in question etc... hes absolutely right on the formula for a wheelie thing and hes done a really good job of explaining it... I'm sure a lot of people will learn stuff through this thread.

    Nice work @Shibboleth!
    • Agree Agree x 2
  15. The numbers you are playing with are the wrong ones. You should be playing with this one > v = u + at
  16. [cough] torque [cough]
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  17. But where does "a" come from? It's the amount of torque that the rear wheel is able to apply to the road.
  18. You were already given the a ~ 11.72
  19. Those numbers seem to be simply the fastest that a cb400 _could_ accelerate before lifting the front wheel off the ground. _Could_ a CB400 do that? I don't know, but the 1.7s 0-70km/h time sounds a little bit suss to me.
  20. Your wrong...

    you base your conclusion entirely on the fact that you have no control over the launch..

    ie. that which routinely sees the dumping of the clutch at the lights and its only dumb luck as to whether the outcome is
    a] the engine bogs down and the bike goes nowhere
    b] all your forward urges head uncontrollably skywards..