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Old Nov 21st 2008, 03:54 PM   #1
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Parabolic trajectory??

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..

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) slowing its lateral velocity and distorting its path into elliptical (Iím talking about a negligible amount, to the point of splitting hairs but letís 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 donít notice.

Thanks for reading and would value responses.

Keith Jump
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Old Nov 21st 2008, 04:50 PM   #2
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You are right, however you said
But let us be exact,
from it I would consider air resistance at least. (it is very signigicant).
But yes, the movement would be an ellipse if there was no air friction (with air friction I'm not sure though).
Do you know about the 1st Law of Kepler?
From wikipedia
"The orbit of every planet is an ellipse with the sun at a focus."
Consider the planet as the projectile and the sun as the Earth and you're done.
If the problem is too hard just let the Universe solve it.
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