Different excercise from Ether's:
Using the second derivative of Ether's equation x(t), the acceleration versus time is:
a(t) = -D*e^(-B*t)
To find out when there's no more acceleration, set a(t) = 0 -- since ln(0) is negative infinity, we basically see that the minibot is always technically accelerating. HOWEVER we can see when it gets "close" to its max. speed by substituting in a very small number for a(t). In my case I chose 0.01 m/s^2 since that's small enough to become "unnoticable" IMO.
Sovling the equation for t, to see
when the minibot is no longer accelerating gives us
t = ln(0.01/-D)*(1/-B)
Then we can plug that 't' back into the original equation to figure out the height at which the minibot is no longer accelerating.
Presuming:
- 0.40" Wheel Diameter
- 2.35lb Weight
- 0.8225lbs of friction (anecdotally determined)
- 2 motors, drive driven to the wheels
- No slip
The time of acceleration = 0.86 seconds
The distance of acceleration = 5.57 feet