You need to wonder and think a little more. It is good for your brain. The negative sign in the equation that I had mentioned earlier shows that for a satellite potential energy is always larger than kinetic energy
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The negative value is not an algebraic sign. In gravitational equations, Potential energy is shown as negative and kinematic energy as positive as the sum for any given object should be constant (U+K=C). Kinematic energy is always positive so potential energy must be negative for the sum to remain constant. One transfers to the other.
Example :
In case our baseline is earth's surface, a 1kg mass, 1m above earth surface potential energy is -mgh=-9.81 j
Now the same mass at 2m above surface has a potential energy of -19.62 j. Hopefully you are not claiming that the one at 1m height has a higher potential energy. You need to compare the absolute values. ABS(19.62)>ABS(9.81)
What you are talking about?
Potential energy itself, can be summed with any constant.
I don't remember your formula, but if it was assumed that the potential energy is zero at infinity(meaning that your constant is zero), then you would always derive potential energy as a negative number! You can sum your potential energy with a constant=+1e999999 as well. Then you would always derive potential energy as a positive number.
To break it down for you:
F=-grad(U) >> U can be derived as:
-integral of F.dl + U0
for two masses: F=-Gm1m2/r^2, hence
U=-Gm1m2/r+U0
Now, if as r goes to infinity, we assume U=0, then U0=0. But, it is not the only solution. You can also assume that as r=radius of Earth, U=0, then U0=Gm1m2/(radius of the earth)
what matters is:
delta E = delta K + delta U
assuming no friction, then delta E=0
Now if r1>r2 then U(r1) >U(r2). and U(ground)= a constant less than both U(r1), and U(r2).
so, deltaU(from ground to r1) > deltaU(from ground to r2)
Consequently, you would need to have more kinetic energy change if you want to send it to r1 compared to r2.
Example :
In case our baseline is earth's surface, a 1kg mass, 1m above earth surface potential energy is -mgh=-9.81 j
Now the same mass at 2m above surface has a potential energy of -19.62 j. Hopefully you are not claiming that the one at 1m height has a higher potential energy. You need to compare the absolute values. ABS(19.62)>ABS(9.81)
You are very lucky that you are not in my vicinity. Other wise, you would have received some physical punishment
if you assume that U(ground)=0, then delta U(from ground to h) = U(h) = +mgh
U(2)>U(1)>0