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

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?

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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.

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So speed is only affected by air friction? As long as motor torque is greater than "slope torque", scooter can reach any speed it is capable of? But in real world I can reach 10 mph using P1 power on θ1 slope, and 15 mph on same P1 power and θ2= 0 (flat road), so I think there's something missing in this formula: I need a formula linking speed to P and θ, or to torque and θ. –  jumpjack Sep 1 '12 at 10:48
/ot How do I add pictures and formulas to my posts? –  jumpjack Sep 1 '12 at 10:49
The equation I've given tells you the maximum slope the scooter can climb i.e. above this slope the motor isn't powerful enough to move the scooter at all. Calculating the maximum speed as a function of slope would be a lot harder as there isn't a simple formula that links maximum speed to the torque - it's a complicated function of air resistance and mechanical friction. –  John Rennie Sep 1 '12 at 11:12
To add pictures just find some suitable image, either as a file on your PC or as a URL, and then click the "Add picture" icon. To draw diagrams like the one above I use Google Drive (drive.google.com) then screenshot the diagram, paste it into Paintshop Pro and save it as a GIF. –  John Rennie Sep 1 '12 at 11:15
So ignoring frictions a vehicle can reach same speed on flat and on a slope?!? –  jumpjack Sep 2 '12 at 12:14