1. **Stating the problem:** Rex Things throws a vase vertically upward with an initial velocity of $26.2$ m/s. We need to determine the maximum height the vase reaches above its starting point. 2. **Relevant physics principle:** When an object is thrown upwards, its velocity decreases due to gravity until it reaches zero at the maximum height. 3. **Known values:** Initial velocity $v_0 = 26.2$ m/s, acceleration due to gravity $g = 9.8$ m/s$^2$ 4. **Using the kinematic equation:** $$ v^2 = v_0^2 - 2g h $$ where $v$ is the final velocity (0 m/s at max height), $h$ is the height gained. 5. **Plugging in values:** $$ 0 = (26.2)^2 - 2 \times 9.8 \times h $$ 6. **Solving for $h$:** $$ 2 \times 9.8 \times h = (26.2)^2 $$ $$ h = \frac{(26.2)^2}{2 \times 9.8} $$ 7. **Calculating:** $(26.2)^2 = 686.44$ $$ h = \frac{686.44}{19.6} = 35.03 $$ m 8. **Conclusion:** The vase rises approximately **35.03 meters** above its initial height.