A rocket is fired straight up from the Earth at 6.0 km/s.

Step 1. $2gR = 1.254 \times 10^{8}\ \text{m}^2/\text{s}^2$ and $v^2 = 3.6 \times 10^{7}\ \text{m}^2/\text{s}^2$.

Step 2. $2gR - v^2 = 1.254 \times 10^{8}\ \text{m}^2/\text{s}^2 - 3.6 \times 10^{7}\ \text{m}^2/\text{s}^2 = 8.94 \times 10^{7}\ \text{m}^2/\text{s}^2$, so $h = 6.4 \times 10^{6}\ \text{m} \times \dfrac{3.6 \times 10^{7}\ \text{m}^2/\text{s}^2}{8.94 \times 10^{7}\ \text{m}^2/\text{s}^2} = 2.576 \times 10^{6}\ \text{m}$.

Step 3. Compare. $\dfrac{v^2}{2g} = 1.837 \times 10^{6}\ \text{m}$.