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I am trying to numerically prove that a motorcycle can do a wheelie from its engine power. However, when I use different methods, I get different results... please help connect the dots !

I see two ways to analyze the point when a motorcycle does a wheelie.

Regarding the physics involved, there are helpful posts in this forum such as this one : How does a wheelie work?

From the torques

If I consider just the torques, I see two, centered around the rear wheel:

  • The torque from the gravity. It's a torque that keeps the bike on the ground and it equals to 9.8*m*x where m is the mass of the motorcycle in kg and x the horizontal distance of the center of gravity from the rear axle in meters
  • The torque from the engine. I simply take the maximum engine torque, multiply it by the transmission ratio and loss factor

So there is a tipping point when the torque from the engine is higher than the torque from the gravity, and this lifts the vehicle.

This tipping point gives me a value at a given vehicle speed (due to the characteristics of the engine and the transmission, not the physics of the motorcycle themselves), and indeed the torque at the wheel from the engine is superior to the torque from the gravity. That's fine : it shows the motorcycle, in the considered gear, at the considered speed, has enough torque to lift the bike. Good.

Other method from an article

Then there is this article : https://www.scienceabc.com/pure-sciences/how-bike-wheelie-works-physics-gravitational-normal-force-torque-angular-momentum.html Or the Physics section of the Wheelie article on Wikipedia : https://en.wikipedia.org/wiki/Wheelie#Physics

In these, there is the formula P=mva where :

  • P = the power required to do a wheelie
  • m = mass of the motorcycle
  • v = velocity of the motorcycle
  • a = acceleration required to do a wheelie

And it's here that it makes no sense to me, and my question is : why would the power required to do a wheelie depend of the velocity of the motorcycle ? This contradicts the analysis above with the torques.

So either this article has a mistake, or I don't understand it, or I'm missing some kind of inertia and the torque analysis is not valid.

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why would the power required to do a wheelie depend of the velocity of the motorcycle ?

Because power, torque, and velocity are related. So you can say "it depends on torque", or you can say "it depends on power and velocity" and they mean the same thing.

If we look at the linear model: $$E = Fd$$ $$\frac Et = F\frac dt$$ $$P = F v$$

So to apply a constant force to a moving object requires increased power as the object's speed increases.

If you already have a figure for the torque of the rear wheel, you're done. If instead you have a figure for engine power, you can calculate the torque given speed (and gearing).

For a device that can generate a nearly constant torque, wheelies would have no dependence on speed. But the motorcycle uses different transmission ratios as speed increases. This means the engine cannot produce as much torque at the rear wheel at higher speeds.

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