Is tension equal to the force a person applies horizontally to an inanimate object? Suppose a person is pulling on a rock, such as below:

Should the tension in the string be exactly equal to the force the person applies when the whole system, (person, rope and the rock) are accelerating towards the persons direction? Because according to newton's second law, F being the applied force of the men:
F-T=ma
the force can't be equal to the tension applied, and if this is the case, can someone give me an intuition on how is this possible? Why isn't the full force I am applying to the string being converted into the tension in the string?
Consider the rope to be massless,
Thank you.
 A: It would require an additional force to move the person, but if we consider only the force the person exerts on an inelastic, massless rope, then all force will apply to moving the rock. If the rock is in a frictionless environment, it will begin to accelerate immediately. If friction is involved, it will accelerate when force is greater than friction. If the rope is elastic, then some force will go into tension of the rope. If force is constant then once the rope tension equals force applied then all force will go to accelerating the rock.
A: You are, for some reason, assuming that the force applied by the man is acting on the man. That is your issue. 
If the ground is frictionless then if the man pulls on the rope with a force $F$ to the left, then by N3L the rope pulls on the man with a force $F$ to the right. In this case there is only one horizontal force acting on the man. Therefore, for the man $F=ma$
If there is friction, then the analysis of the forces does not change, but now you have an additional friction force $f$ acting to the left on the man such that $F-f=ma$. 
In either case the applied force magnitude is equal to the tension magnitude, but the applied force by the man is not acting on the man. Don't just plug all forces in the problem into N2L and expect a valid answer. You need to only include the forces acting on the object in question.
