We know that it was launched with a speed v, and at its highest point, the y-velocity is zero, so the speed of v/2 is solely the x-component (which does not change).
The initial x-velocity is thus also v/2, and the magnitude of the initial velocity is v, so the initial y-velocity is given by the Pythagorean theorem:
v_(0y) = sqrt((v)^2 - (v/2)^2) = (vsqrt3)/2
(which are components of a 30^"o"-60^"o"-90^"o" triangle)
The maximum height h attained is given by the kinematics equation
(v_y)^2 = (v_(0y))^2 - 2gh
Plugging in known values, we have
0 = 0.75v^2 - 2(9.81color(white)(l)"m/s"^2)h
Rearranging and simplifying gives
h = (0.75v^2)/(19.62color(white)(l)"m/s"^2)
We can also simply write it with the acceleration g to make things even simpler:
color(blue)(h = (0.75v^2)/(2g)
or
color(blue)(h = (3v^2)/(8g)