To calculate the speed of a projectile at heights of 80m and 100m, focus on the conservation of energy principle. The total mechanical energy, which is the sum of potential energy (PE) and kinetic energy (KE), remains constant throughout the projectile's motion. Use the formula mgh1 + 1/2 mv1^2 = mgh2 + 1/2 mv2^2, setting the lowest point as zero potential energy. By determining the change in potential energy as the projectile rises, you can find the remaining kinetic energy and subsequently the velocities at the specified heights. Understanding these concepts is crucial for solving projectile motion problems effectively.