Change in Gravity Affecting Free Fall

In summary: r' and r'' are a lot easier to write, and "r-dash" and "r-double-dash" are a lot easier to say (than dr/dt and …) :wink:
  • #1
kaikalii
17
0
In high school physics, which I am in, we learn that to find the distance that and object falls can be found with the equation, x=v0t+1/2gt2. We also learn that the force of gravity between two objects can be found by Fg=Gm1m2/r2 and thus acceleration due to gravity can be derived to be g=Gm/r2

This is all well and good on a small scale, but on a large scale, such as something falling from space, the difference in gravity due to change in distance between the objects is too large to be negligible. As an object falls, the force of gravity, and by extension, its acceleration, increases exponentially.

I have tried to derive an equation that gives the distance that an object will fall (to Earth) as a function of time and initial height, ignoring air resistance, using the equations above, but my knowledge of calculus is only so great, and I keep getting stuck not knowing which variable to solve for or use. I am sure that there is an e in there somewhere as continuous compounding would be needed, but I'm not sure.

Could someone please give me an equation that meets these specifications, and if possible, a step-by-step derivation of said equation?
 
Physics news on Phys.org
  • #2
welcome to pf!

hi kaikalii! welcome to pf! :smile:

x=v0t+1/2gt2 comes from x'' = g, which we integrate once to get x' = gt + constant, and again to get x = gt2/2 + (constant)t + constant

if instead we use r'' = -Gm/r2, we multiply both sides by r' to get r'r'' = -Gmr'/r2, integrate that to get 1/2(r')2 = Gm/r + constant, or r'/√(Gm/r + constant) = √2 … i don't think that has an integral in terms of ordinary functions :redface:
 
  • #3
tiny-tim said:
hi kaikalii! welcome to pf! :smile:

x=v0t+1/2gt2 comes from x'' = g, which we integrate once to get x' = gt + constant, and again to get x = gt2/2 + (constant)t + constant

if instead we use r'' = -Gm/r2, we multiply both sides by r' to get r'r'' = -Gmr'/r2, integrate that to get 1/2(r')2 = Gm/r + constant, or r'/√(Gm/r + constant) = √2 … i don't think that has an integral in terms of ordinary functions :redface:

I'm afraid I'm not familiar with the notation: ' that you are using. Does r' mean "r prime" or "the derivative of r"? If it is the later, then does r'' mean "the derivative of the derivative of r"?
 
  • #4
hi kaikalii! :smile:

(just got up :zzz:)

' means derivative, and '' means derivative of derivative (and so on)

r' and r'' are a lot easier to write, and "r-dash" and "r-double-dash" are a lot easier to say (than dr/dt and …) :wink:
 
  • #5


I appreciate your curiosity and efforts in trying to understand the effects of gravity on free fall. The equations you have mentioned are indeed correct for objects falling on a small scale, such as from a height of a few meters. However, as you have correctly pointed out, on a larger scale, the effects of gravity become more complex and cannot be ignored.

To fully understand the behavior of an object falling from space, we need to take into account the effects of both Newton's law of universal gravitation and Einstein's theory of general relativity. This is because as an object falls towards Earth, it is not only affected by the gravitational force of Earth, but also by the gravitational forces of all other objects in the universe.

To derive an equation that takes all of these factors into account, we would need to use advanced mathematical techniques such as differential equations and tensor calculus. These equations are beyond the scope of high school physics, but they have been developed and studied extensively by scientists and mathematicians.

However, for practical purposes, we can use simplified equations to estimate the distance an object will fall on Earth. One such equation is the one you have mentioned, x=v0t+1/2gt2, which is valid for small heights and low velocities. Another commonly used equation is the one you mentioned for finding the force of gravity, Fg=Gm1m2/r2. This equation assumes that the object falling is a point mass, which is not the case for larger objects.

To accurately predict the distance an object will fall from space, we would need to take into account the shape and density of the object, as well as the effects of air resistance. These factors can greatly affect the object's acceleration and therefore, its distance of fall.

In conclusion, while it is admirable that you are trying to derive an equation for free fall on a larger scale, it is important to remember that the behavior of gravity is complex and cannot be fully explained by simple equations. As scientists, we continue to study and understand the laws of gravity and their effects on objects in free fall.
 

1. How does an increase in gravity affect free fall?

An increase in gravity will cause objects to fall faster due to the increased force of gravity pulling them towards the ground. This means that the acceleration due to gravity will also increase, causing objects to reach the ground in a shorter amount of time.

2. Does an object's mass affect its rate of free fall?

No, an object's mass does not affect its rate of free fall. All objects experience the same acceleration due to gravity regardless of their mass. This was famously demonstrated by Galileo when he dropped two objects of different masses from the Leaning Tower of Pisa and observed that they both hit the ground at the same time.

3. How does the location on Earth affect free fall?

The location on Earth can affect free fall due to differences in the strength of gravity. Objects will fall faster in areas with a stronger gravitational pull, such as near the poles, and slower in areas with a weaker gravitational pull, such as near the equator.

4. What is the formula for calculating free fall?

The formula for calculating free fall is distance = 1/2 * acceleration due to gravity * time2. This formula is based on the laws of motion and can be used to calculate the distance an object will fall in a given amount of time.

5. How does air resistance affect free fall?

Air resistance can slow down the rate of free fall for objects falling through the air. This is because the force of air resistance acts in the opposite direction of the force of gravity, slowing down the object's descent. Objects with a larger surface area, such as a feather, will experience more air resistance and fall slower than objects with a smaller surface area, such as a rock.

Similar threads

Replies
3
Views
877
Replies
2
Views
841
Replies
13
Views
1K
Replies
2
Views
910
Replies
30
Views
4K
Replies
2
Views
736
Replies
11
Views
2K
Replies
17
Views
1K
  • Mechanics
Replies
7
Views
2K
Back
Top