The film and the events it depicts are, in many ways, one giant physics problem. Getting to the moon is one thing, but the specific events depicted in Apollo 13 involved that and more.
Physics problems that any moon trip would involve include;
- Course corrections. In the 1960s, or any era, it's almost impossible to make absolutely precise course adjustments in a single attempt. This is due to the limitations of our instruments, calculations, and control, as well as the fact that very small errors can be magnified greatly over the vast distances that a spacecraft travels.
- Calculating and accounting for changes in mass. It's absolutely critical that the mass of the vehicle be known at all times; however this changes if materials such as gases are intentionally vented or lost in space, or if materials on the moon are picked up or dropped off. This also changes the craft's moment of inertia, since the weight is (presumably) redistributed throughout the craft.
Additional problems that the Apollo 13 crew faced include;
- Serious alteration of the craft's inertial properties, to the point that even the ejection of bodily waste was considered a threat to the craft's trajectory.
- The actual ignition of the oxygen tank was due to an electrical problem; electricity and magnetism are aspects of physics.
- Damage to the heat shield. It was uncertain, and almost impossible to tell, if this damage would prevent the command module from being able to reenter Earth's atmosphere at all. This was the same problem that caused the destruction of the space shuttle Columbia.