Conservation law at high speeds

[stevebd1]

While I understand (in principle) the relationship of gravitational potential energy, kinetic energy and the conservation law in a system for an object falling towards a large mass, I'm not so familiar with how this is applies to an object traveling at high speeds. Say in the example of a spaceship, are we supposed to assume that the potential energy is the fuel on board which is slowly converted to kinetic energy as the object builds up in speed (allowing for some 'work done' quantity) so that the conservation law applies as potential energy reduces (fuel being used) kinetic energy increases? Also how does the conservation law apply at ultra-high speeds where the kinetic energy increases exponentially relative to the Lorentz factor?

 

[Dale]

Here is a good page for relativistic rockets.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

 

[Entropia]

The conservation law still applies at high speeds, but there are some additional factors to consider when dealing with objects traveling at high velocities. First, it's important to note that the conservation law applies to the total energy of a system, which includes both kinetic and potential energy. In the case of a spaceship, the potential energy would indeed be the fuel on board, which is converted into kinetic energy as the object accelerates.

However, at high speeds, we must also take into account the effects of relativity. As an object approaches the speed of light, its kinetic energy increases exponentially relative to the Lorentz factor. This means that the amount of energy needed to accelerate the object further also increases exponentially. So while the potential energy (fuel) may be decreasing, the amount of kinetic energy gained may not be as significant due to the increasing energy required for acceleration.

Additionally, at ultra-high speeds, we must also consider the effects of special relativity on the conservation of momentum. The momentum of an object does not increase linearly with its velocity, but rather follows a more complex equation that takes into account the Lorentz factor. This means that the conservation of momentum may not hold true in the same way at ultra-high speeds.

In conclusion, while the conservation law still applies at high speeds, we must take into account the effects of relativity and the increasing energy requirements for acceleration. At ultra-high speeds, the conservation of momentum may also be affected. It is important to consider these factors when studying the behavior of objects traveling at high velocities.