- Summary
- An equation meant for a projectile's path with variable gravity yields a wacky graph. Is there something wrong with the equation?

The typical equation for the height of a projectile on earth after ##t ## seconds is

## h = -4.9t^{2}+vt+c##

where ##v## is the velocity of the projectile and ##c## the initial height.

This is nice and all but what happens if the height is very large? The leading coefficient of the equation above is half of the acceleration due to gravity on earth. The gravitational force between two objects of given masses is a function of the distance between the objects. If a ball is thrown into the air, the higher it gets, the less it slows down due to gravity.

I assume the distance between our projectile and the earth is the height of the projectile. Then the equation becomes

## h = -\frac{Gm_{earth}}{2h^{2}}t^{2}+vt+c ##

My problem is the graph of the equation. In terms of x and y,

## y = -\frac{C}{y^{2}} x^2+vx ##

where v is a constant and C is a constant (too keep the equation general for projectiles on any planet and to avoid doing some arithmetic).

When I graph, with C and v being greater than zero, the positive part of the graph, in the first quadrant, never seems to intersect the x axis. Intuitively I believe the graph would intersect the x axis, showing there is some time for which the projectile lands. If the projectile never lands, the acceleration must be too small to keep the projectile bound. But when I make C larger, the graph just seems to grow, to expand.

So my question is whether my equation makes physical sense considering the graph suggests it doesn't (to me).

Thank you.

## h = -4.9t^{2}+vt+c##

where ##v## is the velocity of the projectile and ##c## the initial height.

This is nice and all but what happens if the height is very large? The leading coefficient of the equation above is half of the acceleration due to gravity on earth. The gravitational force between two objects of given masses is a function of the distance between the objects. If a ball is thrown into the air, the higher it gets, the less it slows down due to gravity.

I assume the distance between our projectile and the earth is the height of the projectile. Then the equation becomes

## h = -\frac{Gm_{earth}}{2h^{2}}t^{2}+vt+c ##

My problem is the graph of the equation. In terms of x and y,

## y = -\frac{C}{y^{2}} x^2+vx ##

where v is a constant and C is a constant (too keep the equation general for projectiles on any planet and to avoid doing some arithmetic).

When I graph, with C and v being greater than zero, the positive part of the graph, in the first quadrant, never seems to intersect the x axis. Intuitively I believe the graph would intersect the x axis, showing there is some time for which the projectile lands. If the projectile never lands, the acceleration must be too small to keep the projectile bound. But when I make C larger, the graph just seems to grow, to expand.

So my question is whether my equation makes physical sense considering the graph suggests it doesn't (to me).

Thank you.

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