Which statement correctly describes the launch angle that yields the maximum horizontal range for a projectile with a fixed initial speed?

Which statement correctly describes the launch angle that yields the maximum horizontal range for a projectile with a fixed initial speed?

• A. The maximum range occurs at 30 degrees
• B. The maximum range occurs at 45 degrees
• C. The maximum range occurs at 60 degrees
• D. The range is maximum at 90 degrees

Answer: (B)

For maximum horizontal range with a fixed launch speed, you want to maximize how long the projectile spends moving forward while it’s in the air. The horizontal range on level ground is given by R = (v0^2 / g) sin(2θ). Here v0 is the initial speed, θ is the launch angle, and g is gravity. Since v0^2/g is fixed, the range depends on sin(2θ). The sine of 2θ peaks at 1 when 2θ = 90°, which means θ = 45°. At 30° or 60°, sin(2θ) is sin(60°) ≈ 0.866, giving shorter range, and at 90° the sine is sin(180°) = 0, yielding no horizontal distance. So the best angle for maximum horizontal range is 45 degrees.

For maximum horizontal range with a fixed launch speed, you want to maximize how long the projectile spends moving forward while it’s in the air. The horizontal range on level ground is given by R = (v0^2 / g) sin(2θ). Here v0 is the initial speed, θ is the launch angle, and g is gravity. Since v0^2/g is fixed, the range depends on sin(2θ). The sine of 2θ peaks at 1 when 2θ = 90°, which means θ = 45°. At 30° or 60°, sin(2θ) is sin(60°) ≈ 0.866, giving shorter range, and at 90° the sine is sin(180°) = 0, yielding no horizontal distance. So the best angle for maximum horizontal range is 45 degrees.