Do the classics still work? and horizons

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In summary, the projectile equations still hold when the initial velocity is very large. The equation compensates for a rounded surface by adding in a term. For a given height, you can calculate the distance to the horizon by using the equation X/h = D/X.
  • #1
Phymath
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in a case were we use a lot of inital velocity, does the projectiles equations
([tex]x = x_0 + v_0t + \frac{1}{2}at^2[/tex]) still hold I mean when its range is so large that it actually will go beyond the horizon? How does the equation compensate for a rounded surface or does it not have to? btw how far is the horizon for an average humans height, how do u figure that out for a given height?
 
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  • #2
I would say cartesian coordinates are not valid when the curvature of the Earth is taken into account into the projectile equations. Surely [tex]\overline{g}[/tex] will not point towards the "-y" direction but "-r" direction, and both x and y coordinates will have a component of [tex]\overline{g}[/tex]

Anyway, you can model such projectile with polar coordinates, no matter what initial velocity has. In the limit in which the radius of curvature of the ground tends to zero (small trajectories), the polar coordinates collapses over cartesian coordinates.
 
  • #3
The projectile would not follow a strictly parabolic curve if you go *very* far out since the acceleration would not be in the same direction. An extreme example would be an orbiting asteroid, but in this case it would follow a circle-ish shape. I think those problems would be a lot easier to solve using work and energy, just a thought.
 
  • #4
Phymath said:
btw how far is the horizon for an average humans height, how do u figure that out for a given height?

Phymath that is a nice question and IIRC the answer is like this.

To a good approx, on a sphere, when you are height h above surface,
the distance X from your eye to horizon is the solution to this:

X/h = D/X, where D is diameter of sphere

So let us say diameter of Earth is 8000 miles and you are 125 miles above surface, so

X/125 = 8000/X

then to solve
X2 = 1000 000
X = 1000 miles

that is, if you are 125 miles high then you can see 1000 miles

or, if you are 1.25 miles high, then you can see 100 miles.

Another way to write formula is

X2 = Dh = 2Rh
where h is height of observer and R is radius of sphere.

the proof is by pythagoras triangle formula and simple picture
and an approx which depends on h being small compared with R.
 
  • #5
thanks man!
 

1. Do classic scientific theories and principles still hold true today?

Yes, many classic scientific theories and principles have been rigorously tested and confirmed through multiple experiments and observations. They continue to serve as the foundation for modern scientific understanding and are constantly being refined and expanded upon.

2. What makes a scientific theory or principle a classic?

A classic scientific theory or principle is one that has stood the test of time and has been widely accepted by the scientific community for a significant period. It is often based on a large body of evidence and has been repeatedly confirmed through experiments and observations.

3. Are classic scientific theories and principles still relevant in today's rapidly advancing world?

Yes, classic scientific theories and principles are still relevant and important in today's world. They provide a solid foundation for scientific research and advancements, and often serve as the basis for new discoveries and theories.

4. How do classic scientific theories and principles contribute to our understanding of the world?

Classic scientific theories and principles help us make sense of the natural world and provide explanations for phenomena that we observe. They also provide a framework for further research and help us make predictions about future observations.

5. Are there any limitations to relying on classic scientific theories and principles?

While classic scientific theories and principles have been extensively tested and confirmed, they are not infallible. As our understanding of the world evolves, new evidence may arise that challenges or expands upon these theories. It is important for scientists to continue to question and test these theories in order to further our understanding of the world.

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