How do I calculate DC motor speed for a given load? Suppose I have a robot of a given mass, and I'm choosing between 2 different wheels and 2 different motors to put on it. For each wheel I have the diameter, and for each motor I know the stall torque and free speed. How would I figure out which motor and wheel combination will make the robot move the fastest?
My calculations show to use the big motor with the big wheels, but the small motor with the small wheels goes faster than the big motor with the small wheels. I am not sure my calculations are correct, I need to know the correct way to work this problem.
 A: Regardless of the size of the wheels, and ignoring air resistance, if the motor is making $P = T(\omega)\;\omega$ power then the acceleration is
$$ a= \frac{T(\omega)\; \omega}{m v} $$
The motor speed is $\omega = \frac{v}{r} $ where $r$ is the wheel radius. If the torque at $\omega=0$ is $T_0$ and the motor speed at $T=0$ is $\omega_0$ then the torque function is
$$ T(\omega) = T_0 \left( 1- \frac{\omega}{\omega_0}\right) $$
The time it takes to reach a certain speed $v$ is
$$ t = \int_0^v \frac{1}{a}\;{\rm d}v $$ 
$$ t = \int_0^v \frac  { m v } { T_0 \left( 1- \frac{v}{\omega_0\,r}\right) \frac{v}{r} } \; {\rm d} v $$
$$ t = \frac{m \omega_0 r^2}{T_0} \ln\left(\frac{\omega_0 r}{\omega_0 r - u} \right) $$
or
$$ v(t) = v_0 \left( 1 - \boldsymbol{e}^{-\frac{T_0}{r} \frac{t}{m v_0} } \right) $$ where $v_0 = \omega_0\,r$ is the theoretical top speed.
So to get to $99$% of top speed you need
$$ t_{99} = \frac{m\, r^2\, \omega_0} {T_0}\; 2\ln(10) $$
From here you plug in your values and see which one has the highest top speed and which one has the highest acceleration (least time).
A: The solution $might$ be dependent on the context.  Mechanical engineers would do power calculation first.  What is prevalent about these motors, torque or power or frequency?
Concerning torque $\tau = r F$, so $F = \tau/r$ and motor with larger torque and smaller wheels give larger force (push), while the car is accelerating.
Concerning power $P = F v = M \omega$ the size of wheels is irrelevant (well at least for the time car is accelerating) and motor with larger product of torque and frequency will do the best, while accelerating.
Concerning frequency $v = \omega r$, larger frequency and bigger wheels give larger speed.
I thing the first two can give you which shall accelerate faster, while the last one which robot shall have higher final speed.
Let's suppose that power cannot be calculated using free speed.  Then you have four possible robots with four possible accelerations and four possible top speeds. Then results depends on the track length! 
A: This look suspiciously like a homework question, so we're allowed to discuss concepts but not just answer your question (or your professor won't be too pleased :-).
Your question doesn't say what the rolling resistance is e.g. what is the wind resistance the robot feels as it starts moving? If there is no rolling resistance both motors will simply accelerate until they reach their free speed, and in that case you want the highest free speed with the largest wheels.
If there is rolling resistance then the motor will accelerate until the rolling resitance is the same as the force generated at the wheels i.e. the torque times the wheel radius, so it's the torque rather than the free speed that matters.
