If a computer is not plugged into AC power for a while, the internal memory battery dies or gets weak. The memory settings can be lost or get corrupted, and cause startup problems. You probably need a new battery, but first zap PRAM. See
Mac PRAM, NVRAM, CUDA/PMU & Battery Tutorial
Besides the sporces in the tutorial you can get one at a local Batteries Plus store FYI, here's the part numbers & prices.
1/2AA 3.6V Lithium, P/N LITHS14250BA, $8.99 + local sales tax
4.5V Alkaline, P/N COMP3LF22BP, $14.99 + local sales tax
Cheers, Tom