What is the ratio of the initial speed of projectile 1 to the initial speed of projectile 2?

The acceleration along y-axis is constant (because the gravitational force is approximately constant).
Thus, you can use equations for uniformly accelerated motion.
For uniformly accelerated motion, the relationship between velocity and distance travelled is :

v2=v02-2a*s

In our concrete problem, we can write:

vy2=v0y2-2g*h

When the body reaches the maximal height its y-component of velocity is zero, so:

hmax=v0y2/2g

If we use the last equation for both of our bodies:

hmax1/hmax2=(v0y12/2g)/(v0y22/2g)

Thus:

2=(v0y1/v0y2)2

Finally:

Sqrt(2)=(v0y1/v0y2)

Or:

1.41=(v0y1/v0y2)