Ran into this differential equation, got stuck

[Galorian]

Trying to make a calculation and I ran into the following diff equation:

r''=a+b/r^2

And I can't seem to remember how one would go about solving it.

b = 1355041.84
r(0) = 1400
r(t) = 239.6
r'(o) = 0
r'(t) = 75.2
a = ?

I'm specifically looking to find the value of the constant a. If someone could drop off a quick solution that would be great!

(Note: this is part of an attempt to calculate the maximum acceleration of a starship from a video game cutscene as it approaches a small(is) object with a powerful gravity field)

 

[HallsofIvy]

What do you mean by "r'(t)= 75.2"? What value of "t"? "a" is just a parameter in the problem that has to be determined by additional information. Perhaps "r'(t)= 75.2" is that "additional information" but I don't understand what it means.

 

[Galorian]

What do you mean by "r'(t)= 75.2"? What value of "t"? "a" is just a parameter in the problem that has to be determined by additional information. Perhaps "r'(t)= 75.2" is that "additional information" but I don't understand what it means.

I don't know how long it took the ship to accelerate since the scene skips to the slingshot.

All I know is the distance traveled (from a starting distance of 1400km to a distance of slightly over 239.6km), the velocity of the ship as it reached the closer radius, the fact that the ship wasn't moving at t=0 and the fact that the gravity acceleration of the miniature sun it was slingshotting around was about 1355041.84/r^2.

That should be enough conditions to calculate a final answer, I just can't recall how to solve that kind of a differential equation.

[EDIT] Note that I'm not calculating the gravity slingshot maneuver that was preformed right after that approach, only the approach itself.

 

[Galorian]

Nevermind, I just realized that I could calculate the whole thing via gravitational potential energy and hilariously enough the figure came out as higher than the speed of the ship itself...

Thanks anyway! ^_^

 

[blue_raver22]

Dear Scientist,

First of all, it's great that you are trying to solve this differential equation and apply it to a real-world scenario. It shows that you have a strong understanding of mathematics and its applications.

To solve this differential equation, we can use the method of separation of variables. This method involves separating the variables on either side of the equation and integrating both sides to find the solution.

In this case, we can rewrite the equation as:

r'' = a + b/r^2

r'' - a = b/r^2

Now, let's integrate both sides with respect to time:

∫r'' dt - ∫a dt = ∫b/r^2 dt

Integrating r'' with respect to time gives us r' and integrating 1/r^2 with respect to time gives us -1/r. The constant of integration for the first integral will be the initial velocity, r'(0), which we know is 0. The constant of integration for the second integral will be the initial position, r(0), which we also know is 1400.

Therefore, our equation becomes:

r' - a t = -b/r + 1400

To solve for a, we can use the given information about r(t) and r'(t):

r(t) = 239.6

r'(t) = 75.2

Substituting these values into our equation, we get:

75.2 - a t = -1355041.84/239.6 + 1400

Solving for a, we get:

a = (75.2 - 1400 + 1355041.84/239.6)/t

a = (1352196.24/239.6 - 1324.8)/t

a = 5634.514/t

Therefore, the value of the constant a is 5634.514 divided by the time t at which you are trying to find the acceleration.

I hope this helps you in your calculations. Keep up the great work!

Best regards,