A small rocket is fired from a launch pad 20 m above the ground with an initial velocity left angle 250 comma 400 comma 500 right angle ​m/s. A cross wind blowing to the north produces an acceleration of the rocket of 2.5 m divided by s squared. Assume the​ x-axis points​ east, the​ y-axis points​ north, the positive​ z-axis is vertical​ (opposite g), and the ground is horizontal.
  
 
d. Determine the maximum height of the object.


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:

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

\[
0 = 

A small rocket is fired from a launch pad 20 m above the ground with an initial velocity left angle 250 comma 400 comma 500 right angle m/s. A cross wind blowing to the north produces an acceleration of the rocket of 2.5 m divided by s squared. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical (opposite g), and the ground is horizontal. Determine the maximum height of the object.

This physics problem explores how to determine the maximum height of a rocket launched from 20 meters above the ground with an initial velocity vector. The problem involves kinematic equations, projectile motion, and understanding the effects of gravity. Suitable for high school students in Grades 11-12.

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