According to DragTimes who tested a Tesla Model S P100D in Ludicrous+ mode at various stages of charge, concluded that 0 to 60mph times get slower as the state of charge(SOC) decreases along with the maximum power output.
Brooks Weisblat who tested out the Model S started with a 10% battery charge, moving to 25%, to 50%, to 75% and then full charge for the test. It was found that at 10% the 0 to 60mph time was 3.9 seconds, at 25% 2.9 seconds, 50% 2.7 seconds, 75% and 100% 2.5 seconds. The maximum power at 10% was 329kW while at 100% 576kW. At 10% SOC the launch control and Ludicrous+ mode were not available. The Ludicrous+ mode was introduced in January 2017 for the P100D allowing 0-60mph times of 2.4 seconds. However, you will need ideal conditions for this time and possibly a sticky dragstrip.
It has also surfaced recently that Tesla has controversially implemented a power limiting algorithm to limit the amount of uses of the Ludicrous+ mode, but this was to stop unnecessary wear and damage to the drivetrain from overuse of the most powerful mode.
These tests basically show that electric vehicles now and possibly in the near future work just like other electrical devices when at the bottom of their charge where they will have the least voltage and the least power output. This means that if you want the most performance from an electric vehicle, then it's best not to drop the overall charge to very low levels ideally, but this is the mindset and rules for electric vehicles in general.
The test is obviously for 0 to 60 times and not general use, but it does provide a great overview of how the Tesla and electric vehicles in general handle their power in accordance with output over time. Its probable that we need to start thinking about our vehicles now like we do our smartphones. Power levels are everything and if you want to get the best performance from our vehicles, essentially keep them topped up at all times.