Final answer:
By using properties of projectile motion and the relationship between the launch speed and the speed at the projectile's maximum height, the launch angle is found by taking the inverse cosine (arccos) of 1/5.
Step-by-step explanation:
To find the launch angle θ₀ when a projectile’s launch speed is five times its speed at maximum height, we can consider the kinematic equations of motion and the properties of projectile motion. At the peak of its trajectory, a projectile has a vertical velocity of 0 m/s. Thus, the speed at the maximum height would only consist of the horizontal component of velocity which remains constant throughout the flight if we ignore air resistance.
Knowing that the speed at the maximum height is the horizontal component of the initial launch speed, and that the launch speed is five times that at the peak, we can write the following equation: v₀ = 5 * v₀ * cos(θ₀). Simplifying, we get cos(θ₀) = 1/5. Hence, θ₀ = cos⁻¹(1/5), which gives us the launch angle.