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Question

You are sitting on the ground 50 meters from a firework display and are currently watching a bottle rocket fly straight up into the sky at a rate of 40 m/s. How fast must you tilt your head up in order to keep the bottle rocket in view, if your head currently makes an angle of π/3 radians with the ground?

Answer 1

Two basic ways to do this problem, both relying on tan(θ)=y/50, where y is the vertical height of the rocket and dy/dt = 40 m/s. Note θ is the angle of elevation of the rocket from you sitting 50 feet from the launch area.

I. Differentiate without solving for θ:

sec²(θ) dθ/dt = (1/50) dy/dt

When θ = π/3, secθ = 2 (30-60-90), where 50 is the short leg.

dθ/dt = (1/50)(40)/2² = 1/5 = 0.2 radians/sec

II. Solve for θ, then differentiate:

θ = arctan(y/50)

dθ/dt = 1/[(y/50)² + 1] (1/50) dy/dt

When θ = π/3, y = 50√3 (y is the long leg of the 30-60-90):

dθ/dt = (1/4)(1/50)(40) = 10/50 = 0.2 rad/sec