Range of projectile launched from a non-rotating spherical planet

[greyhound]

TL;DR Summary

Let's assume we have a non-rotating planet of radius R and we launched a projectile with initial speed v_0 at angle theta to the tangent plane to the surface. Let's further simplify the situation by assuming that we have constant magnitude acceleration of g. What would be the "range" of the projectile? That is, how far away along the surface from the launch point would the object land?

I figured this would be a problem in some classical mechanics book but so far I can't find an answer anywhere. Assume there is no drag or lift, and since the planet is not rotating we don't have to worry about Coriolis effects. I'm working on a solution but I want to see if my work is correct. Surely this is a problem somewhere? Any one got an answer?

 

[erobz]

TL;DR Summary: Let's assume we have a non-rotating planet of radius R and we launched a projectile with initial speed v_0 at angle theta to the tangent plane to the surface. Let's further simplify the situation by assuming that we have constant magnitude acceleration of g. What would be the "range" of the projectile? That is, how far away along the surface from the launch point would the object land?

I figured this would be a problem in some classical mechanics book but so far I can't find an answer anywhere. Assume there is no drag or lift, and since the planet is not rotating we don't have to worry about Coriolis effects. I'm working on a solution but I want to see if my work is correct. Surely this is a problem somewhere? Any one got an answer?

You say constant magnitude of ##g##, but what about direction?

Also, you must show your work if you are to receive help. The reason to withhold your work is?

 

[greyhound]

Basically I'm assuming that gravity points towards the planet's centre i.e. ##\vec{a}=-g\hat{r}##. You know what, let's the situation and assume that the projectile is shot tangentially to the planet's surface. The reason why I haven't posted any work is that I'm still working on it. I just want to check my answer when I finish it. This isn't for homework or anything, I'm just curious about it. I thought it would be a problem in some classical mechanics book but I can't find anything like it anywhere. I'm at work right now, I'll post what work I have later.

I just wanted to see if anyone happened to know the answer. It seems like a doable question and I'm surprised that I can't find the answer. Or perhaps it's not so doable and that's why I can't find the answer. One thing about it is that it's bugging me and I want to work on it.

You say constant magnitude of ##g##, but what about direction?

Also, you must show your work if you are to receive help. The reason to withhold your work is?

 

[erobz]

Basically I'm assuming that gravity points towards the planet's centre i.e. ##\vec{a}=-g\hat{r}##. You know what, let's the situation and assume that the projectile is shot tangentially to the planet's surface. The reason why I haven't posted any work is that I'm still working on it. I just want to check my answer when I finish it. This isn't for homework or anything, I'm just curious about it. I thought it would be a problem in some classical mechanics book but I can't find anything like it anywhere. I'm at work right now, I'll post what work I have later.

I just wanted to see if anyone happened to know the answer. It seems like a doable question and I'm surprised that I can't find the answer. Or perhaps it's not so doable and that's why I can't find the answer. One thing about it is that it's bugging me and I want to work on it.

It seems to me that some initial velocity could send it into orbit?

 

[Haborix]

I think with your simplification of a constant radial force and a good choice of coordinates, the answer follows fairly straightforwardly. I get ##d=\frac{2 v_0^2 \cos(\theta) \sin(\theta)}{g}##.

EDIT: Of course, I could have made all kinds of errors, so my answer may be meaningless without the steps

EDIT2: Yes, many errors. The above is almost certainly wrong. It assumes the angular velocity is a constant.

 

[greyhound]

I think with your simplification of a constant radial force and a good choice of coordinates, the answer follows fairly straightforwardly. I get ##d=\frac{2r_0 v_0^2 \cos(\theta) \sin(\theta)}{g}##.

I think there might be an issue with your equation. It looks like your answer for range would have units of length##^2##.

 

[Haborix]

I think there might be an issue with your equation. It looks like your answer for range would have units of length##^2##.

Yeah, I fixed that, but not quick enough!

 

[greyhound]

Hopefully you can make out the situation below, but that's what I'm looking for. I'm assuming that ##v_0## is small so that it doesn't orbit. I figure the path will fit an ellipse. I'm just wondering what the equation of motion is for this situation.

 

[Haborix]

Hopefully you can make out the situation below, but that's what I'm looking for. I'm assuming that ##v_0## is small so that it doesn't orbit. I figure the path will fit an ellipse. I'm just wondering what the equation of motion is for this situation.
View attachment 343671

I'm taking a look at my answer again. I think I might have gotten all my angles muddled up.

 

[Baluncore]

Without an atmosphere, I would expect the projectile to follow an elliptical path, a partial orbit, with the near focus at the centre of the planet.

 

[erobz]

Hopefully you can make out the situation below, but that's what I'm looking for. I'm assuming that ##v_0## is small so that it doesn't orbit. I figure the path will fit an ellipse. I'm just wondering what the equation of motion is for this situation.
View attachment 343671

I'm getting a system of second order nonlinear ODE's using Newtons Laws in polar coordinates. I'm trying to find ##r(\theta)##. Maybe I'm missing the bigger picture though.

 

[Haborix]

I'm getting a system of second order nonlinear ODE's using Newtons Laws in polar coordinates. I'm trying to find ##r(\theta)##. Maybe I'm missing the bigger picture though.

That's the kind of mess I'm getting as well now that I worked through it more carefully. Even with the simplification on the functional form for gravity, I don't think there will be a nice closed form solution for the range.

 

[erobz]

Close as I (think) can get for the ODE that would yield ##r(t)## (the radial coordinate- not range):

$$ \ddot r - \frac{v_o^2R^2}{r^3} = -g $$

I don't know how to progress on that( analytically).

 

[Haborix]

Close as I (think) can get for the ODE that would yield ##r(t)## (the radial coordinate- not range):

$$ \ddot r - \frac{v_o^2R^2}{r^3} = -g $$

I don't know how to progress on that.

That's what I got. And then you would need to solve for the time from ##r(t_f)=R##. With that time you could integrate the relationship ##r^2\dot{\phi}=R v_o\sin(\theta)## to get the difference in azimuthal angle from the starting position to the final position. Multiplying that angle by ##R## would give you the range.