I got into the Wait Calculation[1] a few weeks back and made a Jupyter notebook out of it. At our current max speed (692,000 km/hr, stated max speed of Parker Space Probe), it would take about 173,000 years to get to that planet. We could instead choose to Wait and grow our tech. By picking a constant annual growth rate and then doing some calculus to find the minimum, we can calculate the shortest possible time it w…
As others have said, E=mv^2/2 only holds at very small fractions of the speed of light. I believe the equation you need is E=mc^2/sqrt(1-v^2/c^2)-mv^2. https://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_ki...
Most scenarios I've explored - low velocity growth rate, nearby planets/stars - the Wait Calculation tells you to stop researching and start launching long before your tech gets to significant fractions of light speed.