Let's not pretend that you don't know what some people think about electric bikes. Now let's ignore those people completely.
If you want to make it easier to ride uphill or against the wind; if you want to be able to ride faster and longer; if you want to avoid traffic when commuting to work; if you want to enjoy a group ride without straining to keep up, then an electric bike could be for you.
Many people would like to cycle more, to work, on day-to-day errands, if only... An electric bike removes "if only" barriers. With an electric bike, you can arrive at work not covered in sweat, and not have to change clothes or shower at work. You can carry heavy loads, such as groceries, without additional strain. It is not unusual for the owner of an electric bike to get more exercise, because they use their car for fewer (or none) of the short-range trips that previously seemed to be too much trouble to do on a bike.
Electric bikes also make cycling more available to people who may have disabilities, fitness issues or motivational issues that make cycling seem too challenging.
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Because of the weight of the battery and motor, you can expect an electric bike to about double the weight of a typical bike -- to around 50 or 60 pounds. The less expensive the battery technology, the more weight the battery will add. (See below.)
In the US, electric bikes will assist you up to 20 mph. Above 20 mph, it's all you and/or gravity. The range is typically from 20 to 25 miles on a fully-charged battery. A lot depends on the factors independent of the technology, such as the weight of the rider, the weight of the cargo, the terrain, the amount of pedaling the rider does, and even the outside temperature.
Typically, electric bikes range in price from $500 to $4000. The lower end of that range, however, is usually not a good investment. It takes good components to make a good bike, and it takes good technology to make an electric bike pay off. A rider who uses an electric bike on short trips and errands instead of driving a car, will be paid back quickly in money saved on gas, repair and maintenance (oil changes, tune-ups, tires, etc.). Charging an electric-bike battery generally costs less than 20 cents worth of electricity (based on average US costs per kWh).