Are parabolic trajectories really accurate for objects in motion?

[nogginkj]

Parabolic trajectories ?

When you throw an object into the air, fire a cannon ball etc. we assume the trajectory to be that of a parabola, but it is in fact an elliptical path (IGNORING WIND RESISTANCE)

Think about it (ignore wind resistance), we assume that the lateral velocity is unchanging and the vertical velocity is a function of the square of the lateral velocity (ie. Y=X^2, Y being vertical velocity and X being lateral velocity. But this assumption is based on the Earth being flat which it is not.

Therefore as the object moves laterally the force of gravity changes direction as the object is always attracted to the centre of the body ie. Earth, and not always in the same direction as would be required for it to be a parabola, (when you plot a parabola using Cartesian co-ordinates the axis never change direction) and distorting its path into elliptical (I’m talking about a negligible amount, to the point of splitting hairs but let’s be mathematically exact and not just approximate!)

Lets put it another way, imagine the Earth being a super concentrated mass only tiny in size (a few feet in diameter say) but having equal mass, you are standing on an imaginary surface 4000 miles from its centre. The forces of gravity acting on you and the object you are about to throw are identical to that of the the Earth we are all familiar with. You now throw the object into the air at an angle. It rises for a while and then falls back down but is allowed to continue through the imaginary surface towards the centre of the tiny earth, it picks up speed, flies around the tiny Earth passing very close, then increases altitude and slows and comes back to the point where it was thrown from, where you catch it or watch it go up past you to go around again. This orbital path is elliptical.

I think we adopted the parabolic trajectory idea many years ago due to the simpler mathematics in calculating trajectories such as cannon balls etc. and we now assume that these trajectories are parabolic. But let us be exact, not just so close to exact that we don’t notice.

Thanks for reading and would value responses.

Keith Jump

 

[D H]

I think we adopted the parabolic trajectory idea many years ago due to the simpler mathematics in calculating trajectories such as cannon balls etc. and we now assume that these trajectories are parabolic. But let us be exact, not just so close to exact that we don’t notice.

Don't be so quick to nay-say simpler mathematics. There's a lot of benefits that arise from using simpler mathematics. To name but two, closed form solutions, and teaching simpler minds (i.e., kids). We don't have kids jump from 1+1=2 to $R_{ab}-\frac 1 2 R_{cd}g^{cd}g_{ab} = \kappa T_{ab}$. To quote the Wicked Witch of the West, “These things must be handled delicately.”

You are barking up the wrong tree when it comes to parabolas versus ellipses. The trajectory of an object near the surface of the Earth indeed is not a parabola. It isn't an ellipse, either. Air drag has a much, much greater effect on the shape of a typical trajectory than does the non-uniformity of gravitational acceleration. A shooter who accounts for the non-uniformity of the Earth's gravitational acceleration but does not account for windage will not make a very good marksman.

You are barking up the wrong tree when in comes to ellipses being exact. Satellites in Earth orbit do not follow ellipses. The Earth is not a perfect sphere. The Earth's oblateness has a marked effect on orbiting satellites. The Sun and Moon perturb the satellite's orbit. Finally, even after accounting for as many of the Earth's multipole gravitational moments as you want, the Sun, the Moon, and all of the planets, you still will not have an exact model of a satellite's motion because Newtonian physics is not quite right. It is only approximately correct. If you want a better model you will need to go to general relativity. Even then, while general relativity is the best model of gravitation that we have to date, it too most likely is not exactly correct. Some future Einstein will eventually find some flaw in general relativity.

 

[jtbell]

But let us be exact, not just so close to exact that we don’t notice.

Then why not skip classical mechanics and use relativistic quantum mechanics instead?

 

[nogginkj]

If I wanted to do that I would not have posted it in 'Classical Physics'

 

[ZapperZ]

If I wanted to do that I would not have posted it in 'Classical Physics'

Fine. Do classical physics. So why did you ignore the points brought up by D H?

Since you are able to make a simplification of no air resistance, why don't you consider the simplification that is done in elementary mechanics? They are as valid as your "no air resistance" simplification as far as as being pedagogical is concerned. Do a simple projectile motion experiment in your lab, and I dare you to show me that you can detect any "elliptical" motion.

We are not teaching students to be proficient in particle projectile. We are teaching students how to use and apply Newton's law for the first time!

Do you also plan to complain about Ohm's Law next?

Zz.

 

[robphy]

I think we adopted the parabolic trajectory idea many years ago due to the simpler mathematics in calculating trajectories such as cannon balls etc. and we now assume that these trajectories are parabolic. But let us be exact, not just so close to exact that we don’t notice.

Implicitly, physical situations are being modeled by a mathematical problem.
How good the model is depends on our how we intend to use the model.

For the idealized flight of a baseball near the surface of the earth, the parabolic model of the trajectory is probably sufficient... in giving us a good prediction of the motion at the length, time, energy, etc... scales of interest... using mathematical techniques and physical intuition available to a high-school or college student. If the model is insufficient, revise it and try again.

Certainly, one may wish to enlighten folks about the limitations of the model and point to a more general model... a bottom up approach. If you pursue that more general model, one may then judge whether the extra effort (in calculation, academic preparation, etc...) was worth it for the results desired. At the other extreme, one could start off at some advanced viewpoint (e.g. Einstein's Field Equations or a symplectic manifold) then specialize to the approximate parabolic model... but that might be overkill for many.

 

[arildno]

This has already been dealt with, 3 years ago here at PF:
The following thread has a link to a published article on this issue,
https://www.physicsforums.com/showthread.php?t=76700&highlight=parabolic+approximation
(The article, from robphy's post, can be found at:
http://arxiv.org/abs/physics/0310049)
My own comment, post 29, contains a few points with regard to that article which is of general interest:
https://www.physicsforums.com/showpost.php?p=581123&postcount=29

Simplified, if we calle the initial speed V0, the radius of the Earth R, the acceleration due to gravity g, ignoring other classical effects like air resistance, non-sphericity of the Earth, non -uniform density of the Earth and other suchlike phenomena, the parabolic approximation is excellent, as long as $V_{0}$ is a lot less than $\sqrt{gR}$