You are the physics expert for a professional stuntman. The stunt you have designed includes a ramp that is angled at 30°. You expect that the stuntman will be traveling at 18.0 m/s when he leaves the ramp. How far must the landing ramp be positioned, and what is the maximum height of the flames he will be jumping over?

Asuuming the landing ramp is at the same elevation at the takeoff ramp,

Range = (V^2)/g * sin 60 = 38.2 m

maximum height = (V sin30)^2/(2 g) = V^2/8g = 4.1 m.

To determine the distance the landing ramp must be positioned and the maximum height of the flames, we can use the principles of projectile motion and trigonometry.

1. Finding the time of flight:
Since the motion is symmetrical, we can find the time it takes for the stuntman to reach the maximum height of his trajectory. The time of flight (t) can be calculated using the formula:
t = (2 * V * sin(θ)) / g
where:
V = initial velocity = 18.0 m/s
θ = angle of the ramp = 30°
g = acceleration due to gravity = 9.8 m/s²

Substituting the known values into the formula:
t = (2 * 18.0 * sin(30°)) / 9.8
t ≈ 1.08 seconds

2. Finding the horizontal distance traveled:
The horizontal distance traveled by the stuntman can be calculated using the formula:
d = Vx * t
where:
Vx = horizontal component of the velocity
Vx = V * cos(θ)

Substituting the known values into the formula:
Vx = 18.0 * cos(30°)
Vx ≈ 15.588 m/s

d = 15.588 * 1.08
d ≈ 16.82 meters

Therefore, the landing ramp must be positioned approximately 16.82 meters away from the edge of the ramp.

3. Finding the maximum height of the flames:
The maximum height of the flames will depend on the vertical component of the stuntman's trajectory. We can calculate this using the formula:
h = Vy * t - (1/2) * g * t²
where:
Vy = vertical component of the velocity
Vy = V * sin(θ)

Substituting the known values into the formula:
Vy = 18.0 * sin(30°)
Vy ≈ 9.0 m/s

h = (9.0 * 1.08) - (0.5 * 9.8 * 1.08²)
h ≈ 4.862 meters

Therefore, the maximum height of the flames the stuntman will be jumping over is approximately 4.862 meters.