How far apart are two initially $1000m$ separated bodies that fall to earth from infinity, when the closest body reaches earth?
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Imagine two bodies (very, very small compared to the size of the earth), 1(km) separated from each other (in the direction of the earth) at infinity. After they reach earth after an infinite long trip how far will they be separated due to the tidal force? Can you use the tidal force to calculate the distance?. This force which is the difference in gravitational attraction on the bodies, which varies with distance, is zero at infinity and very small if the first plate arrives on earth: g of the first body on arriving at earth (ignoring resistance and assuming the earth's radius r=6,371∗106(m)r=6,371∗106(m):
gfb=MGr2,gfb=MGr2,
where gfbgfb is the acceleration of the first body on earth, MM the mass of the earth,5,97219∗1024(kg)5,97219∗1024(kg) and G the universal gravitational constant, 6,67408∗10−11(m3kg−1sec−2)6,67408∗10−11(m3kg−1sec−2).
Filling in those values we find gfb=39,859∗101340,489∗1012=9,844(msec2)gfb=39,859∗101340,489∗1012=9,844(msec2).
For the second body we, of course, don't know the distance to the earth because this has changed during the trip due to the tidal force (which is what we want to calculate). But let's assume it's just 1000(m)1000(m) more, rr changes from 6,3716,371 to 6,370∗106(m)6,370∗106(m).
In this case
gsb=39,859∗101340,576∗1012=9,823(msec2)gsb=39,859∗101340,576∗1012=9,823(msec2)
As you can see the difference is very small but actually bigger after the bodies made the trip from infinity.
My question: Can we make a calculation involving the (time and distance varying) tidal force acting on the two bodies the whole trip?
It's all just Newtonian mechanics in one dimension, so certainly other means are there for calculating the deviation from 1000(m)1000(m) (if someone knows how, please don't hesitate to answer; I didn't dug into it so deep) when the first body hits the earth.
But my main question was if it could be done with the help of (tiny tiny tiny) tidal forces that influence the distance of the bodies.
P.S. Let's assume (dmckee was so kind to point that out to me; thanks for that; I wrongly assumed that this was clear, which it is of course not!) the initial velocity of the bodies is zero, they're radially alligned with earth, and that their mutual gravitation is zero, for the sake of simplicity. The last assumption is obviously not true (which plays a role at infinity, where the tidal force is zero), and you can even make a new question out of this: What is the ratio of the tidal force between two 1(kg) bodies, 1(km) apart, of which one is on the surface of the earth and the other radially above it? If it turns out that their mutual gravitation is bigger, we must reject this assumption. Only if it's much, much smaller we can make this assumption. But let's assume it for sake of the question.
gfb=MGr2,gfb=MGr2,
where gfbgfb is the acceleration of the first body on earth, MM the mass of the earth,5,97219∗1024(kg)5,97219∗1024(kg) and G the universal gravitational constant, 6,67408∗10−11(m3kg−1sec−2)6,67408∗10−11(m3kg−1sec−2).
Filling in those values we find gfb=39,859∗101340,489∗1012=9,844(msec2)gfb=39,859∗101340,489∗1012=9,844(msec2).
For the second body we, of course, don't know the distance to the earth because this has changed during the trip due to the tidal force (which is what we want to calculate). But let's assume it's just 1000(m)1000(m) more, rr changes from 6,3716,371 to 6,370∗106(m)6,370∗106(m).
In this case
gsb=39,859∗101340,576∗1012=9,823(msec2)gsb=39,859∗101340,576∗1012=9,823(msec2)
As you can see the difference is very small but actually bigger after the bodies made the trip from infinity.
My question: Can we make a calculation involving the (time and distance varying) tidal force acting on the two bodies the whole trip?
It's all just Newtonian mechanics in one dimension, so certainly other means are there for calculating the deviation from 1000(m)1000(m) (if someone knows how, please don't hesitate to answer; I didn't dug into it so deep) when the first body hits the earth.
But my main question was if it could be done with the help of (tiny tiny tiny) tidal forces that influence the distance of the bodies.
P.S. Let's assume (dmckee was so kind to point that out to me; thanks for that; I wrongly assumed that this was clear, which it is of course not!) the initial velocity of the bodies is zero, they're radially alligned with earth, and that their mutual gravitation is zero, for the sake of simplicity. The last assumption is obviously not true (which plays a role at infinity, where the tidal force is zero), and you can even make a new question out of this: What is the ratio of the tidal force between two 1(kg) bodies, 1(km) apart, of which one is on the surface of the earth and the other radially above it? If it turns out that their mutual gravitation is bigger, we must reject this assumption. Only if it's much, much smaller we can make this assumption. But let's assume it for sake of the question.
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