Final answer:
The range of a projectile is equal for two angles of projection alpha and beta that sum to 90 degrees, due to the trigonometric identity sin(90° - θ) = cos(θ), which leads to equal range expressions when substituting beta (90° - alpha) into the range equation.
Step-by-step explanation:
Show the Range for Two Complementary Angles
When analyzing the range of a projectile for two angles of projection α (alpha) and β (beta) where α + β = 90°, we'll need to use the standard range equation for projectile motion without air resistance:
R = ((vo²) * sin(2θ)/g), where 'R' is the range, 'vo' is the initial velocity, 'θ' is the initial angle, and 'g' is the acceleration due to gravity (approximately 9.81 m/s²).
If we take two angles α and β that are complementary (adding up to 90°), we can use the identity sin(90° - θ) = cos(θ). Hence, for a given α, β will be 90° - α.
Now, let's calculate the range for α:
Rα = (vo²) * sin(2α)/g
For β:
Rβ = (vo²) * sin(2β)/g
= (vo²) * sin(2(90° - α))/g
= (vo²) * sin(180° - 2α)/g
Since sin(180° - θ) is equal to sin(θ), we have:
Rβ = (vo²) * sin(2α)/g
Thus, Rα = Rβ, meaning the ranges are equal for two complementary angles.