# How Do You Find Velocity When Acceleration Is Not Constant?

In planetary motion, how do you find change in velocity and change in time in relation to a change in distance? (a=GM/x^2)
since you cannot simply use an equation like vf=vi+at, unless "a" is constant, how do you do it?

vanesch
Staff Emeritus
Gold Member
In planetary motion, how do you find change in velocity and change in time in relation to a change in distance? (a=GM/x^2)
since you cannot simply use an equation like vf=vi+at, unless "a" is constant, how do you do it?

It's called "calculus" and was indeed the mathematical problem that Newton needed to solve before he could formulate his mechanics.

velocity is the integral of the acceleration.

jtbell
Mentor
In planetary motion, how do you find change in velocity and change in time in relation to a change in distance? (a=GM/x^2)

Since planetary motion is two-dimensional (actually three-dimensional, but the orbit is confined to a plane if we're dealing with only one planet at a time, and neglecting perturbations from the other planets), you have to solve a pair of coupled differential equations:

$$\frac{d^2 x}{dt^2} = \frac{GMx}{(x^2 + y^2)^{3/2}}$$

$$\frac{d^2 y}{dt^2} = \frac{GMy}{(x^2 + y^2)^{3/2}}$$

Alright, let me make sure I understand that, since I don't know much calculus. Are the equations relating acceleration at distance x and distance y? Also, how do you exactly go about solving the coupled differential equations? It might be a bit over my head right now, but thanks anyways for your help.

jtbell
Mentor
Do you know about vectors yet?

When you solve differential equations, you get x(t) and y(t), that is, formulas (or tables) for x and y at time t. The first derivatives dx/dt and dy/dt give you the x and y components of the velocity at time t. The second derivatives $d^2 x / dt^2$ and $d^2 y / dt^2$ give you the x and y components of the acceleration.

In practice, people usually solve differential equations like this using computer software. You give it the initial values of x and y at t = 0, and it calculates a table of x and y at later times. It calculates each point based on the results for the preceding point, going one step at a time. There are various methods (algorithms) for doing the calculation, with different combinations of simplicity, speed and accuracy: Euler's method, Runge-Kutta methods, etc. You typically learn the details in a numerical-methods course.

Ok that makes a lot of sense, but it means that I am kind of back where I started hehe. I have a program updating calculations for a, v, x, and y in small increments, assuming that a is constant for a tiny amount of time, which seems more or less like Euler's method. I was wondering how exact my answer would be. Would evaluating the differentials produce a different result than if i simply used a=GM/r^2, vf=vi+at, and dx=vit+1/2at^2 to calculate the new values at every time increment?