system
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human
How do we solve problems that don't have constant acceleration?

I understand basic physics and how we solve motion problems using kinematic equations, but these all rely on a constant acceleration. How do we solve problems where this isn't the case, such as a rocket leaving Earth's gravity(since acceleration due to gravity gets smaller the further away you are).
gpt
Calculus! Lots of it.

We solve Newton's second law which is a second-order ordinary differential equation. Solution methods for finding analytical solutions for several sub-classes of such equations are typically covered in a course on differential equations. Such sub-classes of equations include separable equations, exact equations, linear equations, etc. There may also be so-called *integrals of the motion* which may be exploited to reduce the order of the differential equation, which often makes finding a solution much easier. (Exploiting energy conservation is an example of this last method.)

For a force law of the form **F**(**r**) = k(**r** - **a**)/|**r** - **a**|^(3) (e.g., gravity), there are well-known analytical solutions in terms of conic sections, and these solutions are examined in more detail in a first- or second-year mechanics course.

For general force laws, there is no known method to find an analytical solution (and there may not exist such a solution in terms of elementary functions and their integrals). So we have to use numerical methods to solve the equation approximately. There are many such numerical methods, which vary in accuracy, robustness, and complexity. The basic numerical methods (e.g., Euler's forward different method) are covered in a course on differential equations. More specialized methods are covered in a course on numerical analysis.

Finally, sometimes we are not able to find an analytical solution but we don't really care. For many problems we may only want to find the solution "for small angles" (or some other approximation) or we may only want to know some general dynamics about the long-term behavior of the system. Questions like that can often be more easily treated not by appealing to Newton's second law directly. For instance, if a system can be described by a suitable *Lagrangian function*, then we can extract a lot more information, particularly about the fundamental symmetries of the problem.

In high school or introductory physics, a problem like a rocket launched from Earth would most likely be treated using energy conservation. You can perfectly well answer questions like "what launch speed is required to get the rocket to a certain height?" without having to find a full analytical solution for the rocket's trajectory.