Can a battery really bog down gaming?

[black hawk]

My father's laptop has a weak battery, and windows usually gives that notification about changing the battery. Now, whenever i run prototype on it, the laptop just shuts down. It also overheats i believe, but I am really not sure about that. I saw the ati catalyst center (ati 5470), and went to the powerplay option.
If i set maximum performance while power is plugged in, and maximum battery life when using just the battery, the game run smooth while plugged in, but the laptop shuts down after sometime. If i run the game using the battery, the laptop doesn't shut down but the game lags.

Next, if i set maximum performance on both the things, the same story.
Then, if i set maximum performance on battery and maximum battery life while plugged in, the opposite thing happens, but even in maximum performance, the game lags frequently.
Finally, i am also suffering some serious sound issues that this game is so famous of.