Which speed can an electric scooter reach on a given slope?

Question

Electric scooters are always "given" as "working with slopes up to xx%", but what does it mean? Given motor torque and power, scooter+driver weight and wheels diameter, how can I determine at which speed it will run on a given slope, regardless of air friction?

Answer 1

Suppose you have your scooter on some slope:

The mass of the scooter and driver is $m$ and the angle of the slope is $\theta$. The force $F$ down the slope is given by:

$$ F = mg\space sin\theta $$

And if your wheels have a radius of $r$ then this force creates a torque:

$$ \tau = r \space mg\space sin\theta $$

The scooter will stop when this torque equals the torque generated by the electric motor in your scooter.

Effect of the slope on maximum speed

When the scooter is moving at some speed $v$ the drag on it will come partly from mechanical friction and partly from air resistance. in general the drag will be a complicated function of speed, so I'll just write the drag as a function of velocity $D(v)$. If you know the torque of the electric motor, $\tau$, then the maximum velocity will be given by:

$$ \frac{\tau}{r} = D(v_{max}) $$

where $r$ is the radius of the wheels. If you know the function $D(v)$ you can solve for $v_{max}$.

Now suppose you are on a slope of angle $\theta$ as shown above, then there will be a force due to gravity of $mg\space sin\theta$. The maximum velocity is now given by:

$$ \frac{\tau}{r} - mg\space sin\theta = D(v_{max}) $$

As above, if you know the form of the function $D(v)$ you can solve for $v_{max}$ to find the maximum speed on the slope. The maximum velocity on the slope will obviously be less than the maximum velocity on the flat.