Use the initial projectile speed relation and then compare the two speeds of projectiles with identical mass to calculate the speed of the second projectile.

Given data:

The maximum height of projectile 1 is\(h = 2.6\;{\rm{cm}}\).

The maximum height of the second projectile 1 is \({h_2} = 5.2\;{\rm{cm}}\).

The formula of initial projectile speed is given by the following equation:

\(v = \frac{{m + M}}{m}\sqrt {2gh} \)

Here,m and M are the mass of the first and second projectile, respectively.

The fraction of projectile speed is given by the following:

\(\begin{array}{c}\frac{{{v_2}}}{{{v_1}}} = \frac{{\frac{{m + M}}{m}\sqrt {2g{h_2}} }}{{\frac{{m + M}}{m}\sqrt {2gh} }}\\\frac{{{v_2}}}{{{v_1}}} = \frac{{\sqrt {{h_2}} }}{{\sqrt h }}\end{array}\)

On plugging the values in the above equation,

\(\begin{array}{l}\frac{{{v_2}}}{{{v_1}}} = \frac{{\sqrt {5.2\;{\rm{cm}}} }}{{\sqrt {2.6\;{\rm{cm}}} }}\\\frac{{{v_2}}}{{{v_1}}} = 1.414\\{v_2} = 1.414{v_1}\end{array}\).

Thus, \({v_2} = 1.414{v_1}\) is the speed of the second projectile.