To determine the maximum height of the rocket, we need to focus on its vertical motion (along the $z$-axis). Let's break this down step-by-step:

Step 1: Initial Data

We are given the following information:

• The initial position of the rocket: $\vec{r_0} = \left\langle 0, 0, 20 \right\rangle$ m (since the rocket is launched from 20 m above the ground).
• The initial velocity of the rocket: $\vec{v_0} = \left\langle 250, 400, 500 \right\rangle$ m/s.
• The acceleration due to gravity acts downward with $g = -9.8 \, \text{m/s}^2$ along the $z$-axis.
• There is no vertical wind, so the vertical acceleration remains only the gravitational effect.

Step 2: Equations of Motion for $z$-axis (Vertical Motion)

The vertical component of the motion is described by the kinematic equations under constant acceleration:

$v_z(t) = v_{z0} + a_z t$

$z(t) = z_0 + v_{z0} t + \frac{1}{2} a_z t^2$

Where:

• $v_z(t)$ is the vertical velocity at time $t$.
• $v_{z0} = 500 \, \text{m/s}$ is the initial vertical velocity.
• $a_z = -9.8 \, \text{m/s}^2$ is the vertical acceleration (due to gravity).
• $z_0 = 20 \, \text{m}$ is the initial height of the rocket.

Step 3: Maximum Height Condition

The maximum height occurs when the vertical velocity $v_z(t)$ becomes zero. So, set $v_z(t) = 0$ and solve for the time $t_{\text{max}}$ at which this occurs:

$0 = v_{z0} + a_z t_{\text{max}}$

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