To solve this problem, we can use the principles of projectile motion. We'll consider the motion of the apple in two parts: the upward motion and the downward motion.
First, let's find the initial velocity with which the apple is thrown upward. We know that the final height above the rooftop is 2.00 m and the total height of the building is 30.0 m. Therefore, the apple's initial position would be 30.0 m + 2.00 m = 32.0 m above the ground.
We can use the equation for vertical displacement during upward motion:
Δy = V₀t - (1/2)gt²,
where: Δy = vertical displacement (32.0 m) V₀ = initial vertical velocity (unknown) t = time taken to reach the highest point (also the time taken to fall back down) g = acceleration due to gravity (approximately -9.8 m/s²)
At the highest point, the apple momentarily comes to rest, so its vertical velocity becomes 0. We can use this fact to find the time it takes to reach the highest point.
We can rewrite the equation as:
0 = V₀ - gt,
Solving for t:
t = V₀ / g.
Now, we can substitute this value of t into the equation for displacement to solve for V₀:
32.0 m = V₀ * (V₀ / g) - (1/2) * (-9.8 m/s²) * (V₀ / g)².
Simplifying the equation:
32.0 m = V₀² / g + 4.9 m/s² * (V₀² / g²).
Combining like terms:
32.0 m = (1 + 4.9) * (V₀² / g).
Simplifying further:
32.0 m = 5.9 * (V₀² / g).
Now, solve for V₀:
V₀² = (32.0 m * g) / 5.9.
Taking the square root:
V₀ = sqrt((32.0 m * g) / 5.9).
Substituting the value of acceleration due to gravity:
V₀ = sqrt((32.0 m * 9.8 m/s²) / 5.9).
Calculating V₀:
V₀ ≈ 16.17 m/s.
Therefore, the apple is thrown upward with an initial speed of approximately 16.17 m/s.
Now, let's find the time it takes for the apple to reach the ground. Since the apple is thrown vertically upward and lands back on the ground, the total time of flight will be twice the time it takes to reach the highest point.
Using the equation:
t = V₀ / g,
we can find the time it takes to reach the highest point:
t = 16.17 m/s / 9.8 m/s²,
t ≈ 1.65 s.
The total time of flight is twice this value:
2 * 1.65 s ≈ 3.30 s.
Therefore, the apple takes approximately 3.30 seconds to reach the ground.