**Introduction**

Using some of the physics and battery science that were introduced in the previous two articles, we are now in a position to measure the energy use and maximum range of a given EBike motor and battery combination.

The first step is to estimate how much force is required to move a person on a **non-powered bicycle**. This force, called vehicle resistance is the force required to keep the bike moving at steady speed on a level road. If you were pushing a bike at steady speed on a level road (with the bicyclist not pedaling) then the vehicle resistance is just the force you would apply with your hands to the bike as you pushed it.

**Total Vehicle Resistance**

Total vehicle resistance has two components: it’s the sum of the rolling resistance and aerodynamic drag. Rolling resistance is just the force needed to overcome tire friction at very low speed. Aerodynamic drag is the wind force due to the forward motion thru still air. It you’ve ever stuck your hand out of an auto window and felt the wind force against it you’ve felt aerodynamic drag.

Rolling resistance for bicycles (and other rubber-tire vehicles) is just equal to the weight of the vehicle times a constant, known as the rolling resistance coefficient, which depends on the tire design, inflation pressure, and road surface roughness. Aerodynamic drag is approximately equal to a constant times the square of the velocity. This constant depends on the aerodynamic characteristics of the bike and rider.

At very slow speed where there is little wind resistance (aerodynamic drag), resistance is mostly rolling resistance. For a bicycle with properly inflated tires, the specific rolling resistance is roughly 1% of vehicle weight. If we estimate that the weight of the loaded bike and rider at 220 pounds, the rolling resistance is just 1% of 220 pounds, or 2.2 lb.

The aerodynamic drag must be added to rolling resistance to get the total resistance and energy used. Unfortunately bicycles have rather high aerodynamic drag. The typical resistance force for bicycles is:

**F = 2.2 + 16.5 v2 pounds-force (v in miles/hour)**

As speeds increase, the aerodynamic drag begins to dominate (since it’s proportional to v2). A typical bicycle speed on the level is about 12.5 mi/hr. At this moderate speed, aerodynamic resistance is 2.3 pounds, slightly more than the rolling resistance, and the total resistance goes up to 4.7 lb.

**Mechanical Energy**

Total vehicle resistance is also the mechanical energy required to travel a given distance. Now that we know the force required to push an EBike along, we can calculate the work required, or energy expended. Remember from the previous article, that work, W = f * d, where f is force and d is distance.

There are 5,280 feet in a mile. Therefore, the work required to travel one mile would be 4.7 lb X 5,280 feet, or 24,816 foot-pounds. For this to be useful as a measure of battery capacity, we will need to convert this to Watt-seconds (joules). The conversion factor can be found from a wide range of sources: 1 foot-pound = 1.356 watt-seconds.

Therefore, in the example above, 24,816 ft-lbs X 1.356 watt-seconds/ft-lb = 33,650 watt-seconds. Since there are 3,600 seconds in an hour, in the above example, 33,650 watt-seconds of mechanical energy converts to 9.4 watt-hours.

**Battery Capacity**

To compute the battery’s maximum range, we want to know how much of the battery’s total energy is consumed by traveling one mile. For an electrochemical battery, the total energy available before 100% discharge (assuming 100% efficiency, which, of course, is an inexact over- simplification) is the product of the battery voltage times the ampere-hours rating, which is the total current flow available. For example a 36-volt **EBike battery** that has an 8 ampere-hour rating has a total energy content of 8 X 36 = 288 watt-hours.

Thus, in moving our Electric Bike one mile, we have used 9.4/288 or 3.2% of the battery capacity. Further, based on the existing battery capacity, we can predict a maximum range of 100/3.2 or 30.8 miles. These same calculations can be repeated or modified for different situations, such as rider weight, speed, grade, etc. While some of these calculations may seem a little difficult to grasp at first, actually, the mathematics is quite simple, involving primarily multiplication and division.

**Conclusion**

I hope I have been able to break down here some of the concepts involved in understanding the energy use and maximum range of EBikes. Becoming educated in the physics and mathematics of electric Bikes can pay dividends in the long-run, saving time and money, eliminating confusion, and providing a means for making quantitative comparisons of the performance claims of competing EBike manufacturers.

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