A rocket is launched at an angle of 60.0° above the horizontal with an initial speed of 99 m/s. The rocket moves for 3.00 s along its initial line of motion with an acceleration of 32.0 m/s2. At this time, its engines fail and the rocket proceeds to move as a projectile.

The question is find the max altitude reached by the rocket , find its total time and the horizontal range.

To analyze the rocket's motion, we need to break it into two parts: the initial motion along its launch line and the subsequent projectile motion.

Let's start with the initial motion along the launch line. The rocket's initial velocity can be broken down into horizontal and vertical components.

Horizontal component: v₀x = v₀ * cos(θ)
v₀x = 99 m/s * cos(60°)
v₀x ≈ 99 m/s * 0.5
v₀x ≈ 49.5 m/s

Vertical component: v₀y = v₀ * sin(θ)
v₀y = 99 m/s * sin(60°)
v₀y ≈ 99 m/s * 0.866
v₀y ≈ 85.43 m/s

The rocket moves for 3.00 seconds with an acceleration of 32.0 m/s². Since this acceleration is in the same direction as the initial velocity, we have constant acceleration.

Using the formula: Δx = v₀x * t + 0.5 * a * t²

Δx = 49.5 m/s * 3.00 s + 0.5 * (32.0 m/s²) * (3.00 s)²
Δx ≈ 148.5 m + 0.5 * 32.0 m/s² * 9.00 s²
Δx ≈ 148.5 m + 144.0 m
Δx ≈ 292.5 m

Now, the rocket enters the projectile motion phase where the engines fail and it moves under gravity alone.

Since the rocket's initial velocity in the vertical direction is 85.43 m/s and there is no vertical acceleration (assuming no air resistance), we can use the formula: Δy = v₀y * t

Δy = 85.43 m/s * 3.00 s
Δy ≈ 256.29 m

To determine the maximum height reached by the rocket during the projectile motion phase, we can use the formula: Δy = v₀y² / (2 * g)

Δy = (85.43 m/s)² / (2 * 9.8 m/s²)
Δy ≈ 731.87 m

The total vertical distance traveled by the rocket will be the sum of the initial vertical displacement and the maximum height reached:

Total vertical distance = Δy (initial motion) + Δy (projectile motion)
Total vertical distance ≈ 256.29 m + 731.87 m
Total vertical distance ≈ 988.16 m

Therefore, the rocket will travel approximately 988.16 meters vertically.

and the question is?