Why is velocity zero at a neutral point in a gravitational field? It is said that at neutral points in a gravitational field the net force on a moving mass becomes zero, which means the mass should be moving with constant velocity-and not zero! However, certain textbooks make use of this paradox to solve problems. Is it explicable?

 A: The book calculated the minimum speed required to achieve the goal of reaching the larger mass. But perhaps a more precise way of stating this is:


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*Any speed larger than the calculated one will cause the projectile to definitely reach the larger mass.

*Any speed smaller than the calculated one will cause the projectile to definitely not reach the larger mass.
They calculated the transitional speed.
Now it is arguable whether or not the projectile, launched with the calculated speed, would actually reach the other sphere. I suppose in the idealized mathematical case the projectile would reach the unstable equilibrium in an infinite amount of time, and just hang there. But realistically, (i) you're not going to have perfect spheres, know precisely the distances, etc. in order to set up such a physical system, and (ii) no person in their right mind would actually use this minimum speed to get the projectile to the other sphere; they'd have a bit of wiggle room.
A: There is no paradox. Textbooks that say $F = 0$ implies $v = 0$ are simply wrong, you are right in saying that $F = 0$ only implies uniform motion. Can you show an example of a textbook claiming that?
Edit after the picture:
You've got a lot of answers already explaining why the calculation is correct. Maybe this will help too: 
An analogous problem would be, to which speed do you have to accelerate a ball, such that it can go up a slope and then roll down a second slope, with a flat top between the two slopes. The flat top implies that no forces act on the ball if it reaches that point. This does not imply, that for whatever incoming velocity, the ball will be at rest at the top but simply that the ball won't accelerate or decelerate, once it reaches the top.
Now let's look at the case where the ball is exactly at rest after it reaches the flat top. The flat top is at a height of $h$. The ball has velocity $v$. Taking our point of zero potential energy at the bottom of slope 1, we can simply solve this problem by use of energy conservation: Initially the ball will have a kinetic energy of $\frac{1}{2}mv^2$ and no potential energy. Coming to exactly at rest at the flat top it will have no kinetic energy and a potential energy of $mgh$. Since no energy can get lost, we simply solve for $v = \sqrt{2gh}$. This is the velocity the ball needs to have at the start of slope 1 such that when its at the top it has exactly zero velocity. Now, since no force is acting upon the ball at the top, for any velocity greater than $v = \sqrt{2gh}$, the ball will keep rolling, since no force is acting upon it. Since it can just keep rolling until it reaches slope 2, where the force of gravity will do what it does best and accelerate the ball down slope 2, for any initial speed GREATER THAN $v = \sqrt{2gh}$, the ball will reach the other side of our "mountain".
Hope any of this helps
A: Nothing forces the particle to have no velocity at a so-called neutral point, and indeed the text makes no such claim.
The problem statement is to find the minimum launch speed to reach the other body. This is equivalent to the speed such that the object just coasts past the neutral point with essentially no velocity, since then it can let the distant object's gravity take over. Any less than this and the projectile will reverse course before reaching the neutral point.
So yes, you can have any velocity you want at the neutral point. But the textbook was not asking what velocities are possible there, but rather what velocity is optimal for a certain situation.
A: The text question you cite is a specific problem where that will be true.  You're correct that when the net force is zero, that doesn't imply that the velocity is zero. 
But in this case you are asked to minimize the launch speed and still reach the other sphere. In that case, the speed is minimized when the object crosses the neutral point with a speed arbitrarily close to zero.  
This is not a conclusion from $F_{net} = 0$, but an assumption based on the problem setup.
