I think I found a mathematical error in the 2015 movie The Martian1. Warning, there will be spoilers.
The final plan they settle on to rescue Mark Watney is to have the Tiayang Shen rendezvous with the Hermes, which will perform a gravity assist off of Earth, sending it back to Mars, where Watney will meet it in orbit to return to planet Earth.
But wait, wouldn't it take the same amount of delta-v for the Tiayang Shen to meet the Hermes as it would for the Tiayang Shen to just go all the way to Mars on the same trajectory? If the rendezvous occurs with the Hermes on a Hohmann transfer trajectory between Earth and Mars, and physics 101 says you need to match velocity and position to rendezvous, then the Tiayang Shen must propel itself to an Earth-Mars Hohmann transfer trajectory as well. Perhaps it joins at the cheapest anomaly along the orbit, but it still needs to enter an orbit which would bring it to Mars, Hermes or not. The only difference is slowing down upon reaching Mars, which Hermes may be capable of with its ion engines, but the Tiayang Shen may not be, given limited delta-v. Not to mention that they were originally planning to aerobrake to the surface of mars anyway, so additional maneuvering fuel used after entering orbit could be as low as zero.
Why would having the Hermes speed up the journey? Why did they need to send it back at all given the Tiayang Shen?