Does the rate of hair growth stay constant, or does it change?

Hair follicles (which produce the actual hair) go through a 3-stage cycle. There is a growth cycle, in which they produce new hair, a resting cycle, and a shedding cycle in which the hair is shed. The follicles are not all synchronised, that is they are not all in the same stage of the cycle.

The hair grows at a fairly constant rate, however in many people the growth stage gets shorter and shorter as they get older, and the hair produced becomes thinner and has no pigment (ie you go grey). This may be why your hair appears to be growing more slowly.

  The follicles are also affected by the season, during some months they are more active than others.

i think i may have read somewhere that short hair grows faster than long hair

You can find information about the hair cycle

“Hair cycle includes anagen (the growing phase), catagen (the intermediate phase) and telogen (the shedding phase)… Anagen lasts between three and seven years in most people.” 

“As people grow older the period of anagen shortens. For example, the hair of someone with a five-year anagen can grow to a length of 60 cm before it enters the shedding phase. If their anagen period drops to three years as they age, their hair will then grow only to shoulder length before it falls out or is brushed out. So when a scanty-haired elderly lady boasts that when she was a girl she could sit on her hair, she may very well be telling you the truth!” 

“Over the years, the number of follicles capable of growing hair declines naturally. The decline is especially noticeable on the top of the head. Some follicles increasingly produce only fine, short non-pigmented hairs that look more like vellus hairs than terminal hairs. In older women, this leads to a general thinning of hair. In men it tends to lead to common baldness. If you look at a bald scalp you will see these fine, poorly pigmented hairs.”

MacTheKnife is absolutely right... Hair growth/loss is related to the production of DHT in hair. It's important to maintain a healthy diet and lifestyle. Try to use good products as well (be mindful of alcohol content in your goop!) Use good products like MiN New York's DHT Salon Defense Cleansing System (min.com). For more info, read this: http://www.askmen.com/fashion/fashiontip_150/181_fashion_advice.html