To find the height of the square-based pyramid, we first need to recall the formulas for the volume \( V \) and the area \( A \) of a square-based pyramid.
The volume \( V \) of a pyramid is given by the formula:
\[ V = \frac{1}{3} A_b h \]
where \( A_b \) is the area of the base and \( h \) is the height of the pyramid.
For a square base, the area \( A_b \) is given in the problem as 50 square yards. Hence, we can write:
\[ V = \frac{1}{3} \cdot 50 \cdot h \]
Given that the volume \( V \) is 250 cubic yards, we can set up the equation:
\[ 250 = \frac{1}{3} \cdot 50 \cdot h \]
To solve for \( h \), we first multiply both sides of the equation by 3 to eliminate the fraction:
\[ 750 = 50h \]
Next, we divide both sides by 50:
\[ h = \frac{750}{50} = 15 \]
Thus, the height of the pyramid is:
\[ \boxed{15} \text{ yards} \]