AssalaamuAlaikum,
JazakAllah to Brother GlobalPeace for the Hadith.
Infact I must admit and apologize for my ignorance to not completely explain the answer.
Beating the child lightly is to order him for prayer. But in case if child misses the prayer then he is not obliged to do Qadha (Qazaa) of that prayer if he is not adult. But if he is adult then he has to offer his Qadhaa prayer if he misses one.
And according to following reference if a person does not get puberty even after the age of 15, then prayer is considered obligatory (Fardh) on him from age 15.
http://islam.tc/ask-imam/view.php?q=5559
Here is a very beautiful explanation:
Some people may wonder why a child of ten may be beaten if he does not offer his prayers. This is simply a method of upbringing and education. A child should be encouraged to pray when he is seven years of age. If he refuses to pray, then at the age of ten, a method more serious than mere order may be employed to get him to pray, so that he would have formed the habit of attending to his prayers regularly when prayer becomes obligatory to him on attaining puberty. At this age, corporal punishment is frequently used by parents to instill some discipline into the child. This must always remain very mild, because the aim is to get the child to understand the seriousness of the matter.
When the child is older, this method must stop, because it will no longer be effective in getting the child to pray regularly. Moreover, when prayer becomes obligatory to a child, he should offer it in discharge of his duty, not because he fears to be beaten. I am afraid those who try to compel people to offer their worship in a certain way often exceed the limits allowed by Islam. The worth of worship is in the fact that it is offered at one’s own behest, in response to God’s command, not for fear of any human authority.
reference:
http://www.ourdialogue.com/f2.htm
Allah knows the best and I know very little.
Wassalaam,
VE