To determine the maximum altitude achieved by the rocket, we will solve the problem in two phases:

Phase 1: Rocket's motion under engine power

During this phase, the rocket accelerates upward with a constant acceleration of $a = 10.00 \, \text{m/s}^2$ until the engine cuts off at an altitude of $h_1 = 0.582 \, \text{km} = 582 \, \text{m}$.

We can use the following kinematic equation to determine the velocity at the point when the engine cuts off:

$v_1^2 = v_0^2 + 2a h_1$

Where:

• $v_0 = 0 \, \text{m/s}$ (initial velocity),
• $a = 10.00 \, \text{m/s}^2$,
• $h_1 = 582 \, \text{m}$.

Solving for $v_1$, the velocity at $h_1$.

Phase 2: Rocket's motion after engine cuts off

After the engine cuts off, the rocket continues to move upward under the influence of gravity alone (with an acceleration of $a = -9.81 \, \text{m/s}^2$) until it reaches its maximum altitude, where its velocity becomes zero.

We will use the following kinematic equation to determine the additional altitude $h_2$ gained after the engine cuts off:

$v_2^2 = v_1^2 + 2a h_2$

Where:

• $v_2 = 0 \, \text{m/s}$ (velocity at the maximum altitude),
• $a = -9.81 \, \text{m/s}^2$,
• $v_1$ is the velocity at the engine cutoff.

The total maximum altitude $h_{\text{max}}$ will then be the sum of the altitude at the engine cutoff and the additional altitude gained:

$h_{\text{max}} = h_1 + h_2$

Let's solve this step by step.The maximum altitude achieved by the rocket is approximately 1,175 meters.

Would you like more details on the steps involved, or do you have any further questions?

Here are 5 related questions for further exploration:

1. How would the maximum altitude change if the rocket's upward acceleration were 12 m/s²?
2. What would be the total time of flight until the rocket reaches its maximum altitude?
3. How high would the rocket go if it had an initial velocity of 20 m/s?
4. How does the time to reach the maximum altitude differ between the powered and unpowered phases?
5. What would happen if the rocket encountered air resistance during its ascent?

Tip: When solving kinematic problems, keeping track of units (like converting km to meters) is crucial to avoid mistakes!