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# The Physics Behind Electric Bikes Through Numbers

Our stock of FX and SIERRA for delivery in May has almost run out, and that is a lot earlier than we anticipated. You guys are loving it and to celebrate that, we are going to give you a short but complete article on the physics governing your ebikes rides.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
with F the sum of the forces in Newtons, m the total mass in kgs and your acceleration in m/s². 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 back wheel 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