# The Physics Behind Electric Bikes Through Numbers

Let’s start with the protagonist itself, the ebike, here we will have a look at the SIERRA in particular. The SIERRA is made of a full carbon frame which has to be structurally strong enough to support the sum of your weight, the ebike’s weight with all its components, and the weight multiplication and shocks due to bumps and other obstacles on the road. In addition to all the typical bicycle equipment (disk brakes, gears, chain pedals, wheels, etc) the SIERRA’s frame also has to carry a motor, a battery, an electronic controller and a computer. These components are all relatively heavy with respect to the frame of the bike but remain light once you add your weight to the equation, even for the skinniest of you.

The SIERRA is equipped with a torque sensor which measures how much pressure you are exerting on the pedals. In turn, it sends a message to the controller which calculates how much power from the motor is immediately required. At the same time, the computer scales the controller’s calculations with respect to the assist level you are set on. Eventually, the controller which is connected to the battery, opens a channel between the battery and the motor and allows high power currents to flow in a certain pattern so as to activate the motor and power your ride.

This then makes you go forward and pick-up speed. We will now focus on what is happening once you have reached a speed of 20km/h and are accelerating by 1km/h per second or reaching 25km/h after 5 seconds on a flat asphalt road. According to the second law of Newton, all the forces acting on the bike and yourself while you move forward are equal to your total mass multiplied by your acceleration. This can be written:

**F=ma**

The forces acting on the bike and yourself while you are moving are:

- Your own, applied through the back wheel of the SIERRA: we will call it Fy and it is measured in Newton
- The motor’s, applied through the backwheel and the chain of the SIERRA: we will call it Fm and it is in Newton
- The aerodynamic drag, due to your movement through the air of the atmosphere, which can be calculated in Newton using the following equation:

D=1/2*Cd*p*V²*A with Cd the coefficient of drag, p the density of the air, V your velocity and A your frontal area. - The rolling resistance of the tyres on the road which can be calculated in Newton using:

Fr=Cf*m*g with Cf the rolling resistance coefficient of the tyres on the road, m your mass in kg and g the gravitational constant g=9.81m/s², m*g is essentially the force in Newton exerted downwards by your weight on the bike.

First of all, we can calculate the drag. In a standard relaxed cycling position on the SIERRA, your frontal area is likely to be 0.6m² and your coefficient of drag: 1.15. The density of the air at sea level is 1.225 kg/m3 and your velocity is 20km/h which is equivalent to 5.6 m/s.

**So we get: D = 0.5 * 1.15 * 1.225 * 5.6² * 0.6 = 13 N.**

The rolling resistance coefficient of bicycle tyres on asphalt is equal to 0.004, assuming you are 75kgs, the weight of the SIERRA being 20kgs, your total weight becomes 95kgs.

**Hence, your rolling resistance is Fr = 0.004*95 = 0.38 N.**

So the sum of the forces can now be written:

**Fy + Fm – D – Fr = m*a**

with m your total weight (95kgs) and a your accelerations in m/s².

Also note that the sign of the forces in the sum depends on the direction these forces are acting. If they act in the direction of the movement, then they are positive, if they act in the opposite direction, they are negative. We can replace the variable with the values we calculated knowing that an acceleration of 1km/h/s is equivalent to 0.28 m/s²:

**Fy + Fm – 13 – 0.38 = 95 * 0.28 which is equivalent to Fy + Fm = 40 N**

We know that the wheels of the SIERRA measure about 0.70m in diameter and therefore 0.35m in radius. We also know that torque is a force multiplied by a distance. Hence, the torque generated by the combined efforts of your legs and the SIERRA’s motor is 40*0.35=14 Nm. We can even get the total power by multiplying this by the angular velocity of the wheel, or the speed at which it rotates. We know that its perimeter is 2 * PI * Radius = 2.2m. As we are going at 5.6m/s, this gives 2.5 rotations of the wheels per second or an angular velocity of 15 rad/s (multiply by 2 * PI).

**Hence, on a flat asphalt road, in order to maintain an acceleration of 1 km/h per second while being at a velocity of 20km/h, the total power needed from you and the SIERRA together is 15*14= 210W.**

As the BOFEILI mid-motor of the SIERRA produces 350W of continuous power and more than 600W of peak power, at this cadence you will only be exploiting a third of the power of the beast. Depending on your assist level, you can either fully provide the 210W through your legs or entirely rely on the SIERRA, it’s your choice and that is the magic of electric cycling!

Do not hesitate to ask questions in the comments, we will be happy to answer them if anything needs to be made clearer!