Final answer:
The optimal launch angle to maximize the range of a water balloon, when air resistance is negligible, is 45°. However, with air resistance, the optimal angle is about 38°. The initial velocity also greatly impacts the range, with higher initial speeds resulting in a longer range.
Step-by-step explanation:
The angle at which to launch a water balloon in order to maximize its range is crucial when considering projectile motion. In the absence of air resistance, the angle that maximizes range is 45°. At this angle, given the same launch speed, a projectile will reach its maximum horizontal distance. This concept stems from the fact that the range of a projectile, for level ground and negligible air resistance, is determined by the formula R = (v²/g) * sin(2θ), which reaches its maximum value when sin(2θ) is at its maximum, that is, when 2θ equals 90° or θ equals 45°. When air resistance is taken into account, the optimal angle is slightly less, approximately 38°. Interestingly, for angles other than 45°, there are two launch angles that will produce the same range, and these two angles will sum to 90°.
The initial velocity of a projectile, denoted as v₀, also plays a significant role in determining the projectile's range. The greater the initial velocity, the greater the range.