7.17.
Wee want to find the velocity where the child will leave the slide. If we assume that he leaves the slide at a height h, then we know that from conservation of energy
or 
. Now, if we sum the forces for the child on the
circular part of the slide and set the sum equal to the mass of the
child times his centripital acceleration, we get
we know that the weight of the child is given by
, where 
is the angle from the ground to
the position of the child measured along the arc of the circle.
Plugging in we find
Now, consider the point when the child leaves the slide. At that point, the Normal force on the child by the slide is zero. Also plugging in our value for the velocity found above, we get
From observing the angle , we can see that 
, so solving for h, we find
Notice that this only holds if the child leaves the slide at the point
where the slide is circular. If 
, then the
velocity will be too large to keep the child on the slide once the
slide becomes circular, so he will then leave the track at h = R