Find the escape speed from Mercury, of mass 3.30 × 10²³ kg and radius 2.44 × 10⁶ m…

Step 1. (a) $GM = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2 \times 3.30 \times 10^{23}\ \text{kg} = 2.20 \times 10^{13}\ \text{m}^3/\text{s}^2$, so $v_e = \sqrt{\dfrac{2 \times 2.20 \times 10^{13}\ \text{m}^3/\text{s}^2}{2.44 \times 10^{6}\ \text{m}}} = 4.25\ \text{km/s}$.

Step 2. (b) $\sqrt{\dfrac{27}{3}} = 3$, so $v_e = 33.6\ \text{km/s}$.