Wheelchair Wheelies

Recently, my mother Sally bought her father a wheelchair.  It reminded me of something I've often wondered about wheelchairs: what would it take to go up on the back two wheels?  You could manage it by accelerating forward rapidly, since the wheels would essentially leave the rest of you behind, but how fast would you need to accelerate?

There are two main parameters we need to consider: y, the vertical distance between the center of mass and the rear wheel axis; and θ, the angle between those two points.  They are related by

where r is the distance between the two points.  Differentiating this twice gives

where dots indicate time derivatives.  We know that the CM's acceleration will come from the forward acceleration of the wheels, and from gravity, so we can write

where a is the forward acceleration.  Combining these gives

This is a pretty complex differential equation.  I couldn't come up with an analytical solution, so I decided to do it numerically.  For that though, we'll need some values.  I found a helpful diagram on a Texas Accessibility Standards page, and picked out the relevant bits (measurements in mm):

I plugged in these numbers, along with a guess for a, and put it through WolframAlpha for a numerical solution.  Tweaking a to find when the wheelchair lifts, I found an acceleration of about 30 m/s^2.  To get a sense of scale, that's about the same acceleration involved in going 0 to 60 mph in 1 second.  That would be pretty tough; I think people who do this trick press back on the seat to help.

I'm still feeling pretty crummy; yesterday I had minor surgery to have my vascular access port removed, so expect slow posting in the near future...