There's always confusion with this topic when it's not well explained. It's all inside "work-energy theorem", which says
$$\Delta E_k = W$$
But we'll make a distinction here: work done by conservative forces and work done by non conservative forces:
$$ \Delta E_k = W_C + W_{NC} $$
And now, we just call "minus potential energy" to the work done by conservative ones
$$W_{C}:= -\Delta E_p$$
We do this for convenience. We can do it, because a conservative force is such taht can be written as a substraction of a certain function $B$ like this:
$$W_C=B(\vec{x_f})-B(\vec{x_0}) $$
We just decide to define $E_p=-B$, so $W_{C}=-\Delta E_p$. We include that minus sign so that we can take it to the LHS:
$$ \Delta E_k = W_C + W_{NC} $$
$$ \Delta E_k = -\Delta E_p + W_{NC} $$
$$ \Delta E_k + \Delta E_p = W_{NC} $$
$$ \Delta E_m = W_{NC} $$
So the increment in mechanical energy is always equal to the work done by non-conservative forces. If there are no non-conservative forces, then $\Delta E_m=0$ and energy is conserved (that's why we call them like that.
(read it slowly and understand it well)
So, having this in mind, I think your confusion arises because of that famous "artificial" negative sign.
There are many formulas, and it's typicall to have a mess. It's all about surnames: $\Delta E_k = W_{Total}$, but $\Delta E_m=W_{NC}$. The subindices are the key.
The force of engines is non-conservative. Hence, their work contributes to total mechanical energy.
Gravity is conservative, so we can work with its potential energy.
If there is no increase of kinetic energy, that means
$0 + \Delta E_p = W_{NC}$
So engines are only increasing potential energy. But that means
$$-W_C = W_{NC}$$
Of course, if there's no gain in KE, no acceleration, there's equilibrium. The work of the engines is compensating the work of gravity.
- Negative work is always positive $\Delta E_p$, by definition.
- More altitude means more $E_p$, you are right. But here energy is not conserved (engines). Normally, increasing height would decrease $E_k$, but we're adding work so taht $E_k$ stays constant.
- $\Delta E_k=0$ implies $W_{Total}=0$. That means gravity is making negative work, and engines are doing positive work (equilibrium). The thing is that potential energy variation is minus gravity's work.
I think is wrong as a reference point at r1 the potential energy at infinity should be infinite.
The potential energy at infinity is only infinite if it takes an infinite amount of work to get to infinity. However, because the gravitational force decreases rapidly with distance, a projectile rapidly reaches a space where the force of gravity is not strong enough to reverse its velocity. Potential energy keeps increasing, but there is a maximum value that is approached asymptotically.
We have actually built space probes that have escaped Earth's gravity and the Sun's gravity. Voyager 1 and 2 have exited the Solar System and are never coming back. If they don't run into anything, they will reach arbitrarily far distances. The only thing stopping them from actually reaching infinity is the infinite time it would take to get there. To reach such a state, it only took the energy in the rocket fuel and some kinetic energy stolen from planets during planetary slingshots.
There are systems with a potential energy that reaches infinity at infinite distance. The spring potential energy $U = \frac{1}{2}kx^2$ is an example. When $x=\infty$, $U=\infty$ because the spring force keeps getting stronger as the spring is stretched.
Best Answer
Simply put, potential energy is the energy an object possesses because of its position. Position, or location, is always relative. Therefore there is no such thing as an exact or absolute position in space and consequently no exact potential energy.
Potential energy must be measured relative to something. Suppose a 1 Kg ball is suspended 1 meter above the surface of the earth. Relative to the surface of the earth it has a potential energy of 9.81 Joules. But suppose we put a 0.5 m high table underneath the ball. Relative to the surface of the table it has a potential energy of 4.9 Joules.
We haven't moved the ball, so which is the real potential energy?